A 11 IDO T=m

c

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NBS

PUBLICATIONS

NBSIR 81-2367

NAT'L INST. OF STAND & TECH

AlllDb 4D73E3

Prediction of the Hydraulic Jump

Location Following A Change of

Slope in A Partially Filled

Drainage Pipe

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Building Technology

Washington, DC 20234

and

Drainage Research Group

Department of Building Technology

Brunei University

UK.

May 1981

Issued November 1981

-QC—

100

.056

81-2367

1981

c. 2

U.S DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BURSAL

OF 8TANDARDS

LIBRARY

NBSIR 81-2367

JAN 8 1982

Ho-t net -CJrc.

PREDICTION OF THE HYDRAULIC JUMP

LOCATION FOLLOWING A CHANGE OF

SLOPE IN A PARTIALLY FILLED

DRAINAGE PIPE

Ciaoo

. US' lo no. %i~t^ZC=ri

m/

c.>

Dr. J. A. Swaffield

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Building Technology

Washington, DC 20234

Drainage Research Group

Department of Building Technology

Brunei University

UK.

May 1981

Issued November 1981

and

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

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director
1 <: )>< .IaVUMT* v,«S’A«KaW VJ

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v, t

.:

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ABSTRACT

The criteria governing the formation of a hydraulic jump in a partially

filled fluid conduit downstream of a slope change are presented together

with the necessary techniques to enable water surface profiles and jump

location to be predicted.

Computer programs designed to model the conditions leading to jump formation

under flow and channel scale conditions compatible with current drainage

system design are presented.

The results of a wide range of test conditions in terms of jump formation

and position downstream of a change in channel slope are presented together

with a set of criteria to be used in evaluating whether a jump will occur

for a given set of design conditions.

iii
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/Of f ice

of Policy Development and Research, Division of Energy Building Technology

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

Report prepared by Dr. J. A. Swaf field, guest research worker at NBS-Stevens

Institute of Technology from Brunei University, U.K.

IV
TABLE OF CONTENTS

Page

SUMMARY iii

PREFACE . iv

NOTATION ix

1.0 INTRODUCTION 1

2.0 THEORETICAL CONSIDERATIONS 3

2.1 Steady Uniform Flow in Open Channels 3

2.2 Gradually Varied Flow in Open Channels 5 2.3 The Hydraulic Jump 8

2.4 Loss Coefficients at Channel Slope Transitions 11

3.0 CALCULATION TECHNIQUES AND PRESENTATION OF RESULTS 12

3.1 Determination of Normal and Critical Depths 12

3.2 Numerical Integration for Surface Profiles 12

3.3 Determination of Jump Location 13

3.4 Presentation of Results 14

3.5 Selection of Input Data 14

4.0 DISCUSSION OF RESULTS 15

5.0 CONCLUSIONS AND FURTHER WORK 20

6.0 REFERENCES 21

APPENDIX 1. JUMP LOCATION IN A 0.075 m DIAMETER PIPE Al.l

APPENDIX 2. JUMP LOCATION, 0.10 m DIAMETER PIPE A2.1

APPENDIX 3. JUMP LOCATION, 0.15 m DIAMETER PIPE A3.1

APPENDIX 4. JUMP LOCATION, 0.075 m WIDE CHANNEL A4.1

APPENDIX 5. JUMP LOCATION, 0.10 m WIDE CHANNEL A5.1

APPENDIX 6. JUMP LOCATION, 0.15 m WIDE CHANNEL A6.1

APPENDIX 7. JUMP FORMATION INDICATOR TABLES FOR CONSTANT MANNING

COEFFICIENT OF 0.015 FOR BOTH CIRCULAR AND

RECTANGULAR CHANNELS A7.1

APPENDIX 8. JUMP FORMATION INDICATOR TABLES FOR 0.15 m PIPE

DIAMETER FOR A RANGE AT MANNING COEFFICIENTS FROM

0.009 to 0.018 A8.1

APPENDIX 9. JUMP LOCATION IN A 0.15 m DIAMETER PIPE AT 1/300

SLOPE CARRYING 6 1/s FOR A RANGE OF MANNING

COEFFICIENTS A9.1

APPENDIX 10. SAMPLE OUTPUT, PROGRAM HYDJUMP A10.1

APPENDIX 11. PROGRAM HYDJUMP All.l

APPENDIX 12. PROGRAM HYDSUM A12.1

v
LIST OF FIGURES

Page

Figure 1. Derivation of Chezy Equation for Steady Uniform Flow .... 22

Figure 2. Relationship Between Specific Energy and Flow Depth 23

Figure 3. Basis of Gradually Varied Flow Depth Profile

Calculations 24

Figure 4. Schematic of Numerical Integration to Determine Water

Surface Profile 25

Figure 5. Control Depths Employed in Water Surface Profile

Integration 26

Figure 6 . Forces Acting at Hydraulic Jump Location 27

Figure 7. Summary of Channel Cross Section Shape Calculations ..... 28

Figure 8 . Water Surface Profile and Associated (F&M) Profiles

Used to Position Hydraulic Jump 29

Figure 9. Boundary Conditions Governing Jump Location 30

Figure 10. Boundary Conditions Employed to Indicate Jump

Formation . 32

Figure 11. Schematic Representation of Jump Boundary Conditions .... 32

Figure 12. Schematic Representation of Jump Position Identification

Technique Employed 33

Figure 13. Variation of Normal and Critical Depth with Flow Rate

and Pipe Slope 34

Figure 14. Variation of (F&M) Term with Flow Rate and Pipe Slope ... 35

Figure 15. Jump Location as a Function of Pipe Slope at 6 1/ s Flow

for a 0.15 Diameter Pipe 36

Figure 16. Jump Location in a Rectangular Channel, Width 0.15 m as

a Function of Channel Slope 37

Figure 17. Variation of Jump Location with Flow Rate and Approach

Pipe Slope, 0.15 m Diameter Channel 3 g Figure 18. Variation of Jump Location with Flow Rate and Approach

Pipe Slope, 0.15 m Width Channel 39

vi
List of Figures (cont.)

Page

Figure 19. Pipe Diameter Effect at Q

= 1 1/s 40

Figure 20. Channel Width Effect at Q

= 1 1/ s 41

Figure 21. Effect of Manning Coefficient on Jump Location. Note

n Constant for Approach and Test Pipe 42

Figure 22. Effect of Manning Coefficient on Jump Location. Note n Constant in Approach Pipe at 0.015, Varied in Test

vii
LIST OF TABLES

Page

Table 1. Summary of Hydraulic Jump Position Results for the 0.1 m Diameter Partially Filled Pipe with an Approach Pipe

Slope of 45° 44

Table 2. Summary of Hydraulic Jump Position Results for the 0.1 m Width Channel with an Approach Channel Slope of 45° 45

Table 3. Schematic of Tabulated Jump Boundary Conditions 46

viii
NOTATION

A Channel cross sectional area

C Chezy constant

D Pipe diameter

E Specific energy

F&M Sum of hydrostatic force, F, and momentum, M, at any flow section

F r Froude number

g Acceleration due to gravity

h Flow depth

h Centroid depth

h c Critical flow depth

hn Normal flow depth

L Jump location distance from pipe entry

m Hydraulic mean depth

n Manning coefficient

P Wetted perimeter

Q Flow rate

S Slope of energy grade line

S Q Channel slope

T Water surface width in channel

V Local average velocity

W Width rectangular channel

Z Elevation channel above some datum

p Fluid density

01 Approach pipe slope

02 Test pipe slope

IX

1 INTRODUCTION

Early studies of the formation of a hydraulic jump in open channel fluid flow

may be traced to the 1820 's and to a large extent were responsible for the

classification of flow regimes in open channels now employed. The major work

on jump formation has been, historically, directed towards the large civil

engineering excavated channels with near horizontal bed slopes and straight

sided cross sectional shapes. Both experimental and analytical treatments

find wide application in such open channel design. Characteristically the

presence of a hydraulic jump is responsible for the necessary water surface

profile discontinuity that must occur if flow is established in a channel at

a velocity compatible with the rapid or supercritical flow regime when the

channel slope and roughness characteristics dictate that subcritical, or tranquil, flow regime conditions should prevail. Obviously such a method of

increasing flow depth also provides a potent energy dissipation and flow

mixing mechanism. These characteristics of the hydraulic jump to some extent

explain the interest in this phenomenon in open channel design. Chow [ 1 ] quotes a wide range of useful applications of the jump mechanism in open

channel erosion and bed uplift by decreasing local velocity and increasing

flow depth. Similarly the mixing properties of the jump can be utilized in

chemical water purification processes as well as in aeration.

A review of the literature on the topic therefore reveals a strong bias

toward large scale applications, with the design criteria being designed

with a view to including rather than excluding the hydraulic jump from open

channel flow.

In drainage design both scale and motivation are diametrically opposed to

the situation described above. The conditions governing the formation of a hydraulic jump translate unchanged with the scale of the system; however in

the design of a drainage system the occurrence of a jump is to be avoided if

possible. In a partially filled drainage pipe the occurrence of a hydraulic

jump could result in a local full bore flow that would interfere with the

venting of the system and could lead to back pressure problems upstream.

Vertical stacks discharging into shallow gradient drains represent a design

condition that could lead to jump formation. Additionally the local full

bore flow could deposit suspended material on the crown of the pipe, leading

after sometime to a possibility of pipe restriction and blockage. The

location of a jump in plumbing systems design is of secondary importance.

In the special case of hospital drainage flow in the UK this has been found

to be a particular problem due to the use of the drainage system to transport

macerated disposable bed pans manufactured from strengthened paper-mache.

Thus in drainage design the hydraulic jump is to be avoided where possible

by design.

The objective of the computer simulation described in this report therefore was to produce a usable computer program that could give guidance on the

design condition likely to produce a hydraulic jump. It was intended that

the computer program would be tested in the work reported for a range of flow

1
conditions and pipe sizes and slopes typical of drainage design. These values

incidentally being far removed from the calculation examples found in the

literature for channels carrying flows measured in MG/Day.

However, as mentioned, the basic prediction methods and the flow criteria

translate across the scale differentials. One problem however does become

obvious and that relates to channel cross section shape. The criteria relating

to jump formation are primarily related to flow depth and velocity and hence,

for a known flow rate, to channel cross section. Similarly the forces acting

involve the calculation of flow centroidal depth. The large body of literature

available is almost entirely restricted to simple, straight sided channel

shapes, and understandably so. The introduction of circular sections, while

not in itself difficult in a numerical computer aided solution, dispenses with

any possibility of deriving analytical solutions for jump location, strength or depth change. Douglas et al. [2], for example present a jump location calculation technique for a simple rectangular channel, with a range of simplifying

assumptions, that is typical of the analytical treatments available [3, 4],

The technique chosen for this study was based on the assumption of gradually

varied flow conditions both upstream and downstream of the hydraulic jump

formed in the channel as a result of the slope change. Gradually varied flow

implies that the flow parameters change sufficiently slowed for the steady,

uniform flow equations, characterized by Chezy and Manning, to be applied to

incremental length sections employing the local depth and velocity values.

The criteria determining the formation of the jump are well documented in

terms of a force and momentum balance at the jump between the upstream and

downstream water surface profiles. This force and momentum balance leads to

the concept of conjugate depths and hence to the strength and depth change

associated with the jump. The gradually varied flow assumptions allow the

water surface profiles to be calculated and hence the jump position may be

calculated

.

This report presents the theoretical background to a computer model based on

the gradually varied flow assumption and the force and momentum equivalence

criteria for jump formation. A listing of the Fortran programs written to

model the problem is presented together with flow charts and data input

descriptions that will allow future use of the developed techniques. Results

for a range of flow and pipe and channel design parameters typical of drainage

systems are included, both as summary tables produced by the programs and in

graphical form to illustrate points and trends of particular interest.

2
2 THEORETICAL CONSIDERATIONS

2.1 Steady, Uniform Flow in Open Channels

Figure 1 illustrates the force balance equation for steady flow in an open

channel or partially filled duct. The common expression of this relationship

is known as the Chezy equation where

V = C /~ES^ (1)

m = hydraulic mean depth A/P, m

S Q = sin 6 duct slope

V = mean velocity, m/

s

C = Chezy constant.

The value of Chezy 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 - 1 m

2 / 3 sj/ 2 n

°

Q

= i. A m

2/3 S'!/

2 n

where Q is the flow rate, m 3

/ s A is the flow cross sectional area, m^

( 2 )

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

Chow [1] suggests values in the range 0.009 to 0.018 for materials commonly

found in building drainage systems.

Equation 2 determines the flow depth under steady, uniform conditions, only one value of h yielding the values of A and m necessary to satisfy the equation.

As this depth is by definition constant downstream, dh/dx =0, it must also be

the terminal depth corresponding to the flow terminal velocity at that channel

slope

.

This depth, hn , is commonly referred to as the normal depth.

The specific energy of the flow may be defined as

E h + vi

2g

( 3 )

where h = local flow depth, m

V = local average flow velocity, m/

s

For non-uniform velocity distributions a kinetic energy coefficient weighting

factor, a, may be introduced as av2/2g; here, the assumption is made of uniform

distribution with a = 1.

3
It may be shown that for a rectangular channel, width W, that there are two

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

V

2 Q

2

E=h+2e8 = h+ 2 2gA

Z

A = hW for rectangular case

Q

2

E = h + 7 2g(hW)

h

3 - Eh2 + Q

2 /2gW2 = 0

For a constant specific energy, E, this equation has three roots; two real, one imaginary. Figure 2 illustrates the alternate depths possible for a constant specific energy, generally characterized as rapid and tranquil flow.

It will be shown that the boundary between these two alternatives depends on flow rate and cross sectional shape.

From equation 3 and Figure 2 it may also be seen that the flow specific energy

has a minimum value below which the given flow conditions cannot exist. In a channel of arbitrary cross sectional shape this value may be determined as follows:

E

dE Q

2 da

dh

= 0 = 1 " gA3 dh

(4)

From Figure 2

dA = T dh (3)

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

From equations (4) and (5) the minimum value of E will occur at a depth value,

h c , that satisfies the expression

4
1 - Q

2 T/gA3 = 0 ( 6 )

This value of h is referred to as the flow critical depth hc.

If the normal flow depth hn exceeds h c then the terminal flow would be termed

subcritical, or tranquil flow. If hn

is less than hc then the flow is termed

rapid or supercritical.

It should be stressed that hc 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. It will

be shown that hydraulic jumps may only be established in mild slope channels.

2.2 Gradually Varied Flow in Open Channels

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 expession, equation 2, for the

local flow depth and rate under assumed steady, uniform flow conditions.

Based on the assumptions above and figure 3, the depth profile may be expressed

as follows:

d

dL

V

2 2g + (Z,

. nQ o s

0d + h } = - {-jjjrr (7)

where (Z 0 - S 0L) is the elevation at distance L along the channel, measured in

the downstream direction; S Q is sin 0, channel bed slope,

hence - 1 + S - = nQ

g dL ° dL Am2/3

( 8 )

and as, Q p VA

dV A + V dA = Q

dL dL

and dA = x from equation 5 it follows that

dh

5
dV = -VdA=_VTdh=_QTdh

dL A dL A dL A2 dL

and substituting in equation ( 8 ) yields

Q^T dh 4. 0 _ dh = j n Q j ^

gA3dL ° dL ' An,2 / 3

1 - Q

2 T/g A

3

dL = { 2/3 2 ^ dh

S 0 - (nQ/Ani 'V

l = / r

1

-f|fs^

7

dh

h

S Q - (nQ/Am ) o

( 10 )

where L is the distance between two known depths hQ) h^ .

Figure 4 illustrates this numerical integration, which may be conveniently

achieved by Simpson's rule.

The numerator and denominator of the function to be integrated in equation 10

may be recognized as the equations determining the critical and normal flow

depths in an open channel.

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

is no change in L for a change in h.

When the term S

Q - (nQ/Am^^)^ is zero uniform flow depth is achieved, i.e.,

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

For uniform cross section channels with constant roughness, n, and slope, S Q , the expression ( 10 ) becomes solely a function of flow depth h.

In order to numerically evaluate (10) it is necessary to define boundary

conditions from which the integration may proceed. It should be stressed that

the integration may be carried out either upstream or downstream from a known

depth point. This ability is central to the use of this technique to determine

the position of a profile discontinuity, such as an hydraulic jump.

Figure 5 illustrates the control depths used in the prediction of the water

surface profiles in the case being investigated, namely the change in slope of an open channel.

6
It is assumed in figure 5 that suitable conditions exist for the formation

of a hydraulic jump in the pipe downstream of the slope change. In order to

predict the water surface profiles therefore three control depths are required

as follows:

1) Downstream boundary at C . It may be assumed that the downstream boundary

is formed by the condition that critical flow depth forms at a free

discharge. Experimental work (3) has shown that the depth at such a discharge is slightly less than critical and that the critical depth

occurs slightly upstream, around 0.7 pipe diameters, however this

assumption is sufficient if the channel considered is long.

Naturally if the normal flow depth is less than critical then the flow

depth is unaffected by the presence of the discharge point and normal depth

is maintained up to the exit at C. In this condition the flow is termed

supercritical and the local wave speed is less than the flow velocity.

For this reason information concerning the presence of the open discharge

cannot be transmitted upstream, hence the maintenance of normal depth.

Calculation of the water surface profile upstream from C show that normal

flow depth is approached very rapidly, perhaps within 10-15 pipe diameters.

For this reason a simplification that is considered justified would be to assume normal flow depth, above critical, in the whole pipe section B to

C. This simplification has a bearing or the calculated position of the

hydraulic jump as will be discussed later.

2) The upstream boundary at B. The upstream boundary is dependent on the

exit conditions from the steep pipe AB, figure 5. If the specific

energy at B in pipe AB is known, together with the flow rate, Q, then it

is possible to predict the depth at B in pipe BC. Alternatively the flow

rate and specific energy at B could be used as input information to the

calculation, dispensing with the pipe length AB water profile integration.

3) The entry condition at A. The entry condition at A determines the flow

profile in AB. Critical depth at the entry point A may be used, the

normal depth in the steep pipe AB will be less than this value and the

profile will take on the shape shown in figure 5, approaching normal

depth at some point along AB.

Alternatively the pipe length AB may be assumed long enough for terminal

flow conditions to be achieved at B. This despenses with the calculation

of the water surface profile in AB as equation (2) may be used to determine

normal supercritical depth at B and equation (3) determines the appropriate

specific energy provided the channel depth-area relationship is known.

The choice of the depth increment value dh in the numerical integration is

based, in this treatment, on the difference between the control depth for

that section and the "target" depth. For example in section C to B the

control depth is the critical depth at C while the target is the normal

flow depth reached at some point between C and B. The depth cannot exceed

this target. Hence an appropriate increment site may be • alculated by

7
dh = (hn

- hc )/N, C to B (ID

where N is some reasonable number in the range 10 to 30, depending on the

desired accuracy and computation time.

Similarly for the section B to C

dh = (hc - hB )/N ( 12 )

For the steep slope A to B the increment would be

dh = (hA- hB )/N (13)

where hg is the normal depth expected for that particular channel and flow

conditions.

2 . 3 Hydraulic Jump

The hydraulic jump is an important example of local nonuniform flow. In

drainage design it is to be avoided as the local depth increase may be sufficient to produce full bore flow and associated back pressure problems. However,

in the wider engineering context the inclusion of a hydraulic jump is often

beneficial as a means of rapidly dissipating flow energy, with consequent

reduction in channel erosion problems, the design of power plant turbine tail races being an example.

There is a considerable body of experimental and analytical literature available

on the formation and characteristics of the hydraulic jump. The majority of

this work is related directly to large civil engineering applications and is

also confined to channels of straight sided cross sectional shape, from

rectangular to trapezoidal. This is understandable as such channel shapes

would be the naturally excavated design. Although laboratory models usually

employ rectangular glass sided open channels to study, for example, jump

formation downstream of a sluice gate, no references were found to the case

of jump formation downstream of a slope change in small diameter partially

filled pipes, namely the building drainage case.

The flow process leading to the formation of a hydraulic jump may be explained

as follows: Assume that flow is established in a mild slope channel at a depth

below critical for that channel and flow rate. As the normal flow depth is

greater than critical, i.e., definition of mild slope, the effect of cumulative

friction losses in the channel will be to increase the depth, with a consequent

decrease in local average velocity. This depth change should continue until

normal flow depth is achieved at some downstream point. This cannot happen via

a gradually varied flow process as the theoretical water surface slope would

have to be vertical as the depth passed through the critical value. Hence a discontinuity, or jump, in the depth profile is required to transfer the flow

from supercritical conditions upstream of the jump to subcritical downstream.

It is important to note therefore that such a discontinuity can only occur from

a depth below critical to a depth above critical.

8
In the steady flow case, namely flow conditions constant with time, the

position of this jump may be determined by a consideration of the forces

acting on the fluid at the jump position and the water surface profiles

upstream and downstream of the jump, figure 6.

Referring to figure 6 and assuming both a near horizontal channel, so that the

fluid weight component may be ignored, and steady flow conditions, application

of the momentum equation yields:

Fhl - F h2 ‘ P Q(V 2 - Vi)

where are the hydrostatic forces at 1 and 2 — — 2 2 hence Pg

A

1 h1 - P g

A2 h

2 = pA2V2 - pA^ (14)

where A is the flow cross sectional area and h is the centroid depth,

as illustrated in figure 7.

Rearranging (14) yields:

(pgAh + pgAh + pAV2

) 2 or

(F+M) x = (F+M) 2 (15)

where M is the momentum term pAV^

This analysis assumes that the jump length may be ignored, this allows the

exclusion of local frictional effects over the pipe section containing the jump.

This point will be returned to in the discussion of the prediction model.

It will be seen that both F and M depend on the flow depth and on the relationship between depth and area and hence centroid position.

The complexity of this expression (15) depends entirely therefore on the form

of the depth to area relationship and on the local water surface profiles

either side of the jump. The (F+M) term is sometimes referred to as the

specific force (1); however this not entirely satisfactory due to the momentum

content, and thereafter it is referred to as the (F & M) term.

Thus the position of the jump may be predicted provided that the flow depth

profiles upstream and downstream are known. A knowledge of local depth plus

the steady flow assumption allows all the terms in equation (15) to be calculated and allows the (F & M) values applicable to each point on either water

9
surface profiles upstream and downstream of the jump to be plotted as shown

in figure 8.

The intersection of the two (F & M) curves fixes the jump position. The two

corresponding depths on the upstream and downstream water surface profiles

are the conjugate depths for the jump and allow the energy loss to be

calculated across the jump.

2 2

AE = {h + X_} - {h + Z_)

2g 2 2 g 1 (16)

The formation of a hydraulic jump downstream of a slope change is not inevitable

however. Figure 9 illustrates five conditions that should be considered in

evaluating the possibility of jump formation.

a) If the flow normal depth, dependent on both pipe slope and roughness, is

less than the critical depth, that is independent of slope or roughness,

then no jump will form.

b) If the flow normal depth is greater than the critical depth and the (F & M)

term at pipe entry is greater than the normal depth (F & M) value, then a jump will form as shown in figure 9. Its position may be determined as described previously.

c) If the pipe is insufficiently large in cross sectional area to maintain a free water surface, i.e., open channel flow, then full bore flow will be

established in the pipe. This case is not treated here beyond its

identification by comparing the normal depth calculated to the available

pipe diameter.

d) A more interesting case occurs if the (F & M) value at pipe entry, i.e.,

at the slope change, is less than the (F & M) value appropriate to the

downstream normal flow depth. In this case the jump effectively forms in

the steep pipe, or it may be regarded as drowned at pipe entry. Analysis

of this case requires the introduction of the mass component down the

steep slope appropriate to the water mass contained between the sections

1 and 2 in figure 6. This introduces the physical length of the jump,

obviously a simple treatment regards the jump as concentrated in one location; however in practice it can have lengths several times its downstream depth. Chow [1] presents an analysis based on empirical jump length

measurements for rectangular channels, however no data is available for

partially filled pipes and for the purposes of this study this case is

merely identified by a comparison of the (F & M) term values as mentioned

above.

e) A trivial case is formed when the length of pipe available downstream of

the slope change is short. Here no jump will form if the length is less

than that necessary for the flow depth to increase to the theoretical jump

conjugate depth value appropriate to the upstream water surface profile.

This case is not illustrated in figure 9.

10
Figure 10 illustrates the tests associated with these boundary conditions. It

will be seen that (F & M) curve vs. slope has a minimum value, hence for particular pairs of values of , and 62 the (F & M) terms may change in their

relative magnitudes.

Figure 11 summarizes these tests into a format that will be employed later in

the generation of tables to indicate jump formation possibility.

2.4 Loss Coefficients for Slope Transitions in Open Channel Flow

No data could be obtained on the loss coefficients for slope transitions in

open channel flow. For this reason the results presented assume no loss at the

test pipe entry. The computer program as written has been designed to include

such a loss coefficient, in the range 0 to 1 , should such data become available

from a future experimental program. The effect of such a loss would be to

increase the flow depth at pipe entry, with a consequent decrease in the (F & M)

term at pipe entry. In turn this would have the effect of generally moving the

jump location upstream towards pipe entry. At the lower approach pipe slopes

this effect could result in the jump appearing to become drowned at pipe entry.

Experimental work is required to verify the model and the predicted effects of

the loss coefficient discussed above.

11
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 - Q

2 T/gA3

and normal flow depth

Y = S Q - (n Q/Am2/3

) 2

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 pipe case or 0 < h < W for the square section case. The process is described below:

The initial interval is bisected and a trial value of h = D/2 or W/2, depending

on geometry, is used to evaluate X, Y above. If the resulting values are positive then the sought after root is less than the trial value just used.

A new trial value is obtained by bisecting the interval 0 to D/2 or W/2 and X

and Y recalculated. If the values obtained remain positive, a further reduction in trial value is obtained by bisection.

If the values of X, Y are negative then the desired root is larger than the

trial value and an increased h value is obtained by bisection between the

upper limit, in this case D or W, and the trial value just employed. This

process may be repeated until the required root is obtained.

Due to the need to include the area depth relationship the solution process

must be iterative. The computation time depends largely on the complexity of

the area-depth function.

3.2 Numerical Integration for Surface Profiles

The integration of the position vs depth profile

1 - QT2 /gA3

S Q - (nQ/Am2/3

) 2

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

h0

12
then if the interval h^-hg is divided into 2 equal increments, the value of X

is given by

X =-Ldh [(F(h ) + 4F(h +dh) + F(h+2dh)]

^ o o o

As the integration moves on the length traversed may be accumulated as L = L + X

at the completion of each integration.

3 . 3 Determination of Jump Position

The water surface profiles and the associated (F & M) curves for the flow

upstream and downstream of the jump are illustrated by figure 8. The jump

position is determined in the following manner within the computer program;

figure 12 illustrates:

1) Choose a small increment Ax less than the smallest AL on either of the

water surface profiles. Note AL varies along each profile as shown in

figure 4.

2) For the water surface profile downstream of the jump, i.e. that calculated

by integration back from the critical depth at pipe discharge, determine

the calculated profile points on either side of the Ax value, measured

from pipe entry, points G, H figure 12.

3) Calculate the (F & M) value at position Ax on the downstream profile,

point J.

4) For the profile calculated by integration downstream from pipe entry

determine whether the corresponding (F & M) values at the known profile

points on either side of Ax are bracketing the trial (F & M) value from

step 3, points K, M. If this is the case the curves intersect in this

increment, if both are greater than the trial value then the curves have

not intersected up to this interval.

5) If the curves have not intersected, increase the search position from

x to x + Ax and repeat steps 2 through 4.

6) If the curves have intersected then the intersection point can be

obtained by solving the two straight line equations representing the

water surface profiles between the two pairs of known points bracketing

the intersection position. This is illustrated by points UV, and XY in

figure 12.

The technique uses the simplifying assumption that the water depth downstream of the jump is at the normal flow level, hence (F & M) for C to B

becomes a known constant. This simplification was extensively used in the

results presented (in figure 12 the resulting downstream depth was H then

became the constant normal depth value).

13
3.4 Presentation of Results

The parameters governing the flow and the location of the hydraulic jump are numerous. The selected test cases, analyzed by program HYDJUMP run on the

NBS CBT Perkin Elmer 732 computer, are summarized below:

1.1) Pipe diameters: 0.075, 0.10, 0.15 m. 1.2) Manning coefficient 0.015.

1.3) Test pipe slopes at all pipe diameters: 1/40, 1/80, 1/100, 1/200

(Additionally slopes 1/150, 1/300, 1/400, 1/600 were run for the 0.15 m diameter pipe only).

1.4) Approach pipe slopes 2°, 4°, 6°, 10°, 20°, 30°, 45°, 60°, 75° and 90°.

1.5) Additionally for the 0.15 m pipe at 1/300 slope, the effect of Manning

coefficient values of 0.009, 0.012, 0.015 and 0.018 on jump position for

the whole range of approach slopes was carried out at one flow rate.

2.1) Rectangular channels, width 0.075, 0.10, 0.15 m. 2.2) Manning coefficient 0.015 only.

2.3) Test pipe slopes 1/40, 1/80, 1/100, 1/200.

2.4) Approach pipe slopes 2°, 4°, 6°, 10°, 20°, 30°, 45°, 60°, 75° and 90°.

The full results are presented in tabular form in a series of appendices.

In addition plotted data is presented to illustrate the main points in the

discussion of the results.

The results indicated the need to determine the boundary conditions in terms of

normal and critical flow depths, pipe diameter and (F & M) values. This led to an additional program HYDSUM, capable of calculating normal and critical depths

as well as (F & M) values. Tables based on this program's output are included

as a method of determining whether a jump will form in the test pipe.

The two main programs, HYDJUMP and HYDSUM are included in an appendix together

with flow charts and input data instructions.

3.5 Selection of Input Data

The choice of input test conditions was governed by the range of values likely

to be found in drainage systems. The pipe diameters chosen, 0.075, 0.10 and

0.15 conform to this criteria as do the pipe gradients used for all test cases,

1/40, 1/80, 1/100, 1/200. The choice of pipe roughness or Manning coefficient

was more difficult, however values in the range 0.009 to 0.018 are recommended

in many texts, i.e. Jaeger [3] and Chow [1].

As previously discussed, the losses at the change of slope that produce the

conditions conducive to hydraulic jump formation have been ignored in this

treatment. No available data on open channel transition loss coefficients for

partially filled pipes or channels could be obtained. The program is capable

of dealing with transition losses however via an input data control variable

provided the loss can be expressed as a factor, 0 to 1.0, of the specific

energy of the flow at pipe entry.

14
4. DISCUSSION

The position of the hydraulic jump in a mild slope channel following a change

in bed slope is determined by the equivalence of the hydrostatic force plus

momentum terms outlined in equations 14 and 15 and figures 6 and 8. If, as a starting assumption, the flow downstream of the jump is assumed to be uniform

and at normal depth, then the jump position appears to be entirely dependent

on the rate of change of the (F & M) term with respect to distance along the

channel. This is demonstrated in figure 12, and the following analysis.

The results for jump location therefore may be discussed in terms of the

parameters governing this gradient, d(F & M)/dL.

From equation 14, that the (F & M) term at any location may be expressed as

(F & M) = pgAh + pAV2

F & M = pgATT + p s?

A (17)

d(F & M) = pgA dh_ + p gh dA -

dL dL dL

p dA

a2 dL

(18)

The form of equation (17) indicates that (F & M) will initially decrease as depth and hence area A increase, however at some depth value the hydrostatic

force term will predominate and the (F & M) value will increase. It therefore

follows that there would theoretically be two intersection, points for the two

(F & M) curves in figure 12, however the intersection closest to the pipe entry

is the valid solution and the technique described in figure 12 will ensure that

this is the solution produced. The form of equation (18) confirms this point,

d(F & M)/dL being initially negative due to the predominance of the momentum

term.

As the rate of decrease of (F & M) is dependent on both depth and rate of change

of depth, it is necessary to incorporate the Manning equation to represent local

friction losses at the equivalent steady, uniform flow conditions. This appears

in the denominator of equation (10) and may be used to explain the movement of

the jump position.

Ah

AL

S ,

. 2/3.2

= o - (n Q/Am )

1 - Q

2 T/gA3

(19)

Changes in S Q , Q, and n that result in an increase in the value of Ah/AL will

indicate an increase in d(F & M)/dL, however this does not automatically imply

that the jump position will move upstream.

15
Figures 13 and 14 illustrate the dependence of flow normal and critical depths

and (F & M) values on flow rate and pipe slope for the 0.15 m diameter pipe

case. It will be seen that as Q increases or S Q decreases, the values of

normal depth and (F & M) increase. This effectively reduces the necessary drop

from the (F & M) value associated with the water surface profile at pipe entry

upstream of the jump, figure 12, and would imply, for a constant d(F & M)/dL, a jump location movement towards pipe entry.

Consider the jump location movement illustrated in figure 15. For constant Q,

D, n and test pipe slope, S 0 = Sin ©2 , the value of Ah/ AL will be constant,

equation (19). Similarly for constant Q, D, n and S 0 the value of normal depth

in the test pipe, figure 13, will also be constant, providing a constant target

(F & M) value.

However as the approach pipe slope, ©i , increases the value of the entry

conditions to the test pipe change, the entry depth decreases and the entry

velocity rises. Thus as 0^ increases the entry value of (F & M) at L=0

increases, this effect for constant flow rate is illustrated in figure 14.

Therefore the intersection point between the (F & M) curves associated with the

water surface profiles upstream and downstream of the jump would be expected to

move downstream as the approach pipe slope increased. This effect is illustrated

in figure 15 for each of the test pipe slopes considered. Figure 16 confirms

this result for the rectangular section channel.

Inherent in the definition of the hydraulic jump location is the criterion that

the depth upstream of the jump is below critical and that downstream is above

critical. This implies that the Froude Number (upstream of the jump), defined

as

is greater than unity upstream and less than unity downstream. This results in

the inability of jump position information to be transmitted upstream as the

wave speed is less than the local flow velocity. As a result of this the depth

downstream of the jump may be thought of as the sole determinant of jump

position for similar flow situations. This effect is illustrated in figure 15

and 1 6

.

Consider the case of constant approach pipe slope 6} and constant flow rate Q,

i.e. vertical lines drawn in the L - sin 6^ plane. In this situation the

(F & M) value at pipe inlet remains constant, however as the test pipe slope, 02

decreases the normal flow depth increases and with it the target (F 4 M) value,

figures 13 and 14. Thus the downstream flow depth controls the jump position,

the jump moving towards the pipe inlet as the test pipe becomes less steep.

Jaeger [3] quotes this depth effect as the sole determinant of jump position in

the case of hydraulic jump formation upstream of an obstruction to the flow.

As indicated in figure 15 jump location may be expressed by an equation of the

form

16
L - Lx

- C ( 21 ) „ . , n n n0.25 K sin(e-0o

)

It was felt that such an empirical formula had little value in this situation

due to the complexity of the boundary conditions already described and the

ability of the computer simulation to predict jump positions for a wide range

of flow conditions in a relatively short time. It would appear from the

available literature, although primarily directed to large civil engineering

applications, that this conclusion is shared by most investigators, as no similar relationship was found. In every case it was recommended that jump

position be calculated by the gradually varied flow analysis presented for

each set of flow conditions.

The downstream depth also controls jump position as indicated by the results

presented in figures 17 and 18. Here for constant S Q = sin 0^ = 1/200 and

constant D and n the jump location moves upstream as the flow rate decreases

from 8 to 1 l/s. Reference to figure 14 at any value of sin 0^ indicates that

the differential between (F & M) at L = 0, i.e. pipe entry value, and the (F & M)

value appropriate to normal depth in the test pipe, decreases as the flow rate

decreases. Hence, with reference to figure 12, the intersection position would

be expected to move upstream, as confirmed for both circular and rectangular

cross section channels by figures 17 and 18.

It will be noted that none of the jump position curves pass through the

origin. This is to be expected as theoretically the jump location would tend

to L = 0 as approach pipe slope 9^ approaches the test pipe slope 02- In addition to this effect some of the curves, namely those representing shallow

test pipe slopes, intersect the sin 0} axis at approach pipe slopes greater

than this minimum. This is due to the drowning of the jumps at pipe entry, as illustrated in figure 9, Case 4, caused by a reversal of the differential

between (F & M) at pipe entry and the (F & M) value associated with the normal

depth flow in the test pipe. As mentioned previously the jump moves upstream

into the approach pipe and a calculation of its position requires a knowledge

of jump length and approach pipe slope. No theoretical calculation technique

for jump length, i.e. the distance between control sections 1 and 2 in figure

6, is available, the data on this variable being entirely empirical and

generally applicable only to straight sided channel shapes.

Tabulated results for the full range of test cases represented by figure 15 to

18 for pipe diameters and channel widths 0.075 to 0.15 m are presented in

Appendices 1 to 6. Tables 1 and 2 present typical examples for the 0.1 m diameter pipe and channel cases. These results for the smaller dimensioned

channels, 0.075 and 0.10 m diameter and width confirm the discussion presented

above. Figures 19 and 20 are representative of these results.

Referring to figures 19 and 20, it will be seen that considerable difficulty was experienced in producing comparable test cases for the three channel dimensions chosen. This is entirely due to the influence of the boundary conditions

introduced in figures 9 to 11. Indeed no jump formation was possible at 1/4(J

for the full range of test cases dealt with. This result prompted the preparation of tabular data based on the boundary conditions illustrated in figure lu

17
that would allow the formation of a jump, but not its location, to be predicted

at considerably less computing time.

Table 3 illustrates the technique. Full tables for the test cases covered are presented in Appendix 7 and 8.

Referring to table 3, assume that the flow rate is Q2 , approach pipe slope is

and test pipe slope is S 2. Pipe diameter is D and the Manning coefficient

is n. The table has been produced by means of equations (2), (6) and (14) and

allows the following boundary checks to be made, corresponding to figure 9.

1) Case 1, compare hnj ^2 to hc2 » if ^n l,2 ^ h°2 , no jump possible.

2) Case 2, compare hn^ 2 to ^c2 » if ^n l 2 ^ h°2 » j umP possible.

3) Case 3, compare hn^ 2 to D if hn^ 2 = D, full bore flow, jump forms but

location not determined.

4) Case 4, compare the normal flow depth value of (F & M) at Q2 and in

the test pipe, (F & M)^2> to the value of the (F & M)* term at pipe entry.

It is assumed that this is equal to the terminal (F & M)* value in the

approach pipe and that the approach pipe was sufficiently long to allow

normal flow depth to be established. Approach pipe slope designated U^.

hence (i) if (F & M)j 2 < (F & M)^, jump possible

(ii) if (F & M)]^ > (F & M)^ 2 > then the jump forms in the approach pipe or

is drowned at pipe entry.

Appendix 8 presents similar tables for the 0.15 diameter test pipe case only

for a range of Manning coefficient values 0.009 to 0.018. This table was used

to determine a suitable test case for an evaluation of the effect of Manning

coefficient on the jump location in a 0.15 m diameter channel. From Appendix 8

employing the method outlined above for table 3, a test case based in b 1/s

flow rate at a pipe slope of 1/300 was seen to produce a hydraulic jump at

each of the values of Manning coefficient between 0.009 and 0.018 chosen.

Appendix 9 contains the results of this test series while figures 21 and 22

summarize the data.

Two test cases were investigated.

1) Manning coefficient constant for both approach pipe and test pipe.

Figure 21 illustrates this case. As is to be expected the jump location

moves downstream as the value of Manning coefficient decreases. This is

in line with Jaeger [3] comments on the importance of downstream depth as the normal flow depth will decrease with decreasing Manning coefficient.

Similarly this result agrees with the description of jump location move- ment based on (F & M) values. As n decreases the normal flow depth (F & M)

value in the test pipe decreases. For the approach pipe the decreasing n

18
also results in a reduced normal depth, however due to the slope of the

(F & M) vs slope curve illustrated by figure 10, this results in an increased value of (F & M) at test pipe entry. This increase in the

differential between the entry and target (F & M) values in the test pipe

effectively impose a downstream movement on to the intersection point and

jump location, figure 12.

2) The Manning coefficient held constant at 0.015 for the approach pipe and

only varied for the test pipe.

The general trend in these results is similar to that described above.

The constant value of n = 0.015 for the approach pipe effectively decreases

the entry (F & M) value for the n = 0.009 and n = 0.012 cases, comparing

figures 21 and 22. Hence the jump position would be expected to move upstream, as is demonstrated by the results. For the n = 0.018 case, the

approach pipe effectively has a reduced n value and hence the terminal

(F & M) value is increased and the jump position should move downstream.

Again the results confirm this.

All the results discussed above were obtained with the assumption that terminal

conditions had been achieved in the approach pipe and that the flow downstream

of the jump could be considered uniform, i.e., the depth was equal to the

normal flow depth defined previously. This assumption was necessary if any

order was to be found in the results produced by the computer program. The

program however was capable of producing jump location for any combination of

pipe lengths, slopes and roughness coefficients.

Appendix 10 presents the full program output for a given set of pipe lengths

and slopes. This output may also be used to justify the normal flow depth

assumption downstream of the jump, as 95 percent of normal depth is achieved in some 20 pipe diameters. The output also presents the water surface profiles in

the approach pipe and upstream of the jump in the test pipe.

Appendix 11 contains the full print out of the program HYDJUMP employed to

generate these results and the control data necessary to vary the format of the

data output and control the assumptions made in the calculations.

Similarly Appendix 12 presents the program HYDSUM employed to produce the

tabular guides to jump formation summarized in table 1.

19
5 CONCLUSIONS AND FURTHER WORK

The objective of this study was the development and testing of a computer program capable of predicting the location of a hydraulic jump

following a change in slope in an open channel or partially filled pipe.

The program HYDJUMP presented has been shown to be capable of jump prediction

based on the assumptions inherent in the application of gradually varied flow

analysis

.

An extensive range of program test conditions were considered within the limits

set by parameter values encountered in building drainage system design.

It was found possible to explain the movement of jump location with such

parameters as flow rate, pipe slope, and roughness in terms of the criteria

determining jump formation in near horizontal channels.

Although a considerable body of literature exists on hydraulic jump formation

in civil engineering style open channels, no comparable study was found employing

the parameter values presented. Similarly the majority of the available

literature referred to straight sided open channels rather than to partially

filled pipes of small diameter. All analytical solutions were restricted to

the simplest rectangular section channels due to the complexity introduced by

the depth to area and centroid position funcitons in non-rectangular sections.

The computer program, as included, has been designed to deal with rectangular or circular section channels. Extension to other shapes requires only that the

shape functions be introduced into the two main subroutines, BOUND and CALC,

and that the range of control values assigned to the control variable SHAPE be

extended accordingly.

It is felt that the next stage in this study should be an experimental phase

designed to compare the predictions from HYDJUMP with laboratory observations,

incorporating modified assumptions where necessary. The loss at the pipe slope

change, although incorporated in the program for future use, has not been

investigated as to do so without laboratory test backing would be of little

value.

As a result of the computer study, several boundary conditions that govern the

formation of a jump were identified. As an aid to future work these boundary

conditions were incorporated into a program HYDSUM, also presented, that

enables the probability of jump formation to be assumed prior to running the more time consuming HYDJUMP to determine its location. The predictions of

HYDSUM could also be used in designing a laboratory test series.

20
6 REFERENCES

.

[1] Chow, V.T., Open Channel Hydraulics, McGraw Hill, 1970.

[2] Douglas, J.F, Gasiorek, J.M. and Swaffield, J.A., Fluid Mechanics,

Pitman 1979.

[3] Jaeger, C., Engineering Fluid Mechanics, Blackie and Sons, London, 1936.

[4] Streeter, V.L. and Wylie, E.B., Fluid Mechanics, McGraw Hill, 1974.

21
At stations 1 and 2 the following equations apply:

Energy equation

Losses = hf = Pi ~ P2 +

pg

as V} = V2 ; steady, uniform flow.

Momentum equation (down slope direction)

( PI ~ P2)A + PgAAx sine - t0 AxP = 0

as dV/dt = 0; steady flow.

.

•

. Pi " P2 + Az = to AxP = h*

Pg PgA

1 ? For turbulent flow to = f ^ pv

h

f = f , V = C / mS n , = sine

2gm o’ om = A/ p C = constant

Figure 1. Derivation of Chezy equation for steady uniform flow. Note no shape restriction on channel.

22
Figure 2.

2

Relationship between specific energy and flow depth, illustrating

alternate rapid or tranquil flow regimes.

23
Gradually varied flow, analysis based on head loss at any

to Manning loss prediction, where

section being equal

Figure 3. Representation of the gradually varied flow depth profile model

24
Note 1. Al increases as calculation proceeds

2. Calculation proceeds downstream from A to give profile hQ h < h c ; h at A taken as intitial condition.

3. Calculation proceeds upstream from X to give profile hc

h < hn ; h at X taken equal to critical depth as initial condition.

A. Value of N dependent on situation, range 10-30.

Figure A. Schematic representation of numerical integration to determine

water surface profile.

23
a a el £ ^ t «. rrv\ \ rv a\ £

a't U = -zero

Note: Water surface profiles approach hn rapidly from an initial condition

h = hc.

Figure 5. Control depths employed in determining water surface profiles

upstream and downstream of a pipe slope change.

26
Fl

F 2 Mi

M2

p8AljU

pgA2 h2

( pAiV]_ )V

i

(pA2 V2 )V2

hj 2 ~ centroid depth = f(channel shape)

Fi 2 “ hydrostatic forces

Mi 2 - momentum crossing cont rol

volume boundaries at 1,2

Jump occurs if (F+M)i = (F+M) 2

.

Depths hi, h2 known as conjugate depths.

Figure 6. Forces acting at the jump location in a horizontal

horizontal channel.

> or near

27
T « \N T

Depth

Area

Surface width T

Perimeter _

= Centroid depth h =

Figure 7. Summary

-5>

h X w

= (D 2 /8) X (6 -sine)

“ = 2.0 x h x CD = h)

2h + w

— D x b /

2

h/2 = h - D/2 + XQ

/ 3 x sin62 - s in 3b/2.\

X

o = 3

X

2 X \4 x (9/2 - X2 x sin V)

J

of duct cross section parameter calculations.

28
L= O.

Prof-Ae ftB ca>cu\*-ted

as &k vjcXue.

ProjA* y'V C<Acu\«-t«cL

(XO kk vcCVue

-J-rom Vv

o U>UY\ (kC "K 0) ^ ^

Jjron-% K c

wttK CK^-K^/vj

Figure 8. Water surface profiles and associated (F+M) profiles employed to

position hydraulic jump.

29
Note: h

c given by: Q^T/gA^ - 1 - 0

2 Note: h

n given by: S Q - (n Q/Am‘'^) = 0

Figure 9. Conditions governing jump formation.

30
Note; if (F+M) a > (F+M) b or c » jump drowned at L=0.

if (F+M)a < (F+M) b or c , jump possible,

if (h ) A

< (h

c

) A

jump impossible.

> D, full backflow.

Figure 10. Tests employed to determine the possibility of jump formation.

31
Figure 11. Schematic representation of jump boundary conditions.

32
P-V-tA

K

Figure 12. Schematic representation of jump position identification technique

employed.

33

I
o

o CO

-c £ x e

34

Figure

13.

Variation

of

normal

depth,

h

n

,

and

critical

depth,

h

c

,

with

flowrate

Q

and

test

pipe

slope

02*

Note

values

independent

of

bi

.
Figure 14. Variation of (F+M) term with pipe slope and flow rate. D - 0.15,

n = 0.015.

35
s'»n

'/

tOO

*/l

SO V

zoo

'/soo

'/w-OO

Ul

36

Figure

15.

Jump

location

asa

function

of

pipe

slopes

0^02

for

constantQ- 6

U/s) D0.15

and

Manning

coefficient =

0.015.
oS

©c

37

Figure

16.

Variation

of

jump

position

with

test

pipe

slope

1/80 -

1/200.

Flow

constant

at Q=

80

1/s W=

0.15

m,n=

0.015.

Rectangular

channel

results
'ooc

03

LP

** 0\ 0

!* £

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<*

A o 0 o v

-5 £ c -X)« O

U>

co

©I <zf

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CO

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006

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05

J-d

c46

<D 17st ,o» . 6J

38

Figure

17.

Variation

of

jump

location

with

flow

rate Q

and

approach

pipe

slope

for

constant D=

0.015

m,n=

0.015

and

1/200

test

pipe

slope.
Figure 18. Jump position in a rectangular channel, width 0.15 m, slope 1/200,

as a function of flow rate and approach pipe slope.

39
Note curves do not pass through L = 0.

Figure 19. Pipe diameter effect at Q = 1 (1/s) 0 2 = sin 1 (0.005), Manning

coefficient = 0.015.

40
Note curves do not pass through L/W = 0.

Figure 20. Channel width effect at Q

= l(l/s) 0£ = sin ^(0.005)

coefficient = 0.015.

• 07 5 m

\ O m

\-o

Manning

41
Figure 21. Effect of Manning coefficient n on jump location in a pipe at 1/300

with a flow rate of 6 (1/s). Pipe 0.15 m diameter.

42
Figure 22. Manning coefficient effect. Note: n constant for approach pipe at

0.015.

43
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44

Table

1.

Jump

location

ina

0.1m

diameter

partially

filled

pipe

at

an

approach

pipe

slope

of

45°.
C

OHM JN

DATA

APPROACH

PIPF

DATA ^ • T" .a •

to to ^ a.

a. • c c • 3

*1 u. Z

to Mi O J

<xX •

til * X * T* UJ U

too • at X

to

X

X o

UJ "3

r

flu • J c

it -a

X 9 u- ©

f/y

X a,

to

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**

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X u* • i© x a X

at

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u

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to a. c < flu uj *• © X <o toat c • UJ to • z V X u. M UJ &

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to

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UJ Z X

to UJ UJ

i_ a. e z JJ uj o z • x z u • x z

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UN ru

©

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ta rj .-» 'VJ -a o "V "SJ o o © © o O O o to o O • Q o © © 9 o o o o 9 9 O o ©

tu 9 © « ® UJ 9 <0 9 9 • A* OJ • 9 9 © o © © a> © © © © V eu © © ©

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to to to to to UJ 9 9 • 9 UJ 9 9 9 UJ UJ 9 9 9 wv »j* t/l \A to 3 © © © © © 3 o *• to o © ©

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to to to to to z X z z e=3 <n *4 n no 4* ISt mm mm

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to to to to to u. VJ

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to to to < tv 4 to to

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to

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to to to to to to to to to to to to to to to to to to to to o © © o © © Q © © © © © © © © © © © © ©

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to to to to to to to to to to to to to to to to 4 4 4 -4 9 9 9 9 9 9 9 O © o o o o o o o © o © o o o © © © © © © © o © © o o © o o 9 9 © o © 9 o © o © 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 « *o «# VI to ro © to to X 43 •r VI to n a to 4

45

lahlo

2.

Jump

location

ina

0.1m

wide

channel

at

an
Table 3 . Schematic of Tabulated Jump Boundary Conditions.

Flow rate Q Qi Q2 Ql D

Pipe diameter

Test pipe

slope S = sin 02

Normal flow

depth, Q, S values.

S

1 s 2

hn12

hn2 2

c f

• • Si hn i2 D

cf

Critical depth

he = f ( Q

)

hci hc 2

hc-^

Flow rate Q Qi Q2 Qi

Test pipe slope (F+M) test pipe normal

depth value

Si (F+M ) 12

• • c f S3 •

Si

• (f+m) i2

Approach pipe (F+M)* terminal

slope U (i.ex test pipe entry)

U

1 (F+M) *i 2

Uo

(F+M) *22

(F+M* i2

46

Q

Q
APPENDIX 1

JUMP LOCATION IN A 0.075 m DIAMETER PIPE,

MANNING COEFFICIENT 0.015, AT SLOPES 1/40, 1/80, 1/100, 1/200

Al.l
COMMON

DATA

APPROACH

PIPE

DATA

TEST

PIPE

DATA

AND

PROGRAM

RESULTS

a

-l o c • s r

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aa

o* t*1 *• o O is®

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X

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r

x «* u, ru

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r • * * z r u. v o x x • u t z z

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0.075

FULL

BORE

FlOR

ESTABLISHED

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TEST

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DlA.

MANN*

SLOPE

HN

TERM.

SLOPE

HC

mm

EMTRY

ENTRY

FmTRY

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DEPTH

ENERGY

ENERGY

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JUMP

JJ1P

L/S

M.

COEfF

ISINI

M.

ENERCY

ISlMl

M.

M.

OfcPTH

ENERGY FM

OEPIH

DEPTH

CHANGE

UP

JUMP

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CHANGE F

N.

PJS.

A1.3 c<

015

0.0698

0.0M7

0.1MI

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FULL

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ESTABLISHED
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SLOPE

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CUFFF

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DEPTH

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CHANGE

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PQ>.

0.»)b

0.«))
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MANN.

SLOPE

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MC

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(SIM)

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DEPTH

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JUMP

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F«H.

POS.

0.07

0.015

0.

1716

0.046

0.274

0.0050

0.073

0.075

FULL

BURE F

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ESTABLISHED
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data

APPK'UCH

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to «3> CL t A. Z £ • 3 £

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A8.9
'
APPENDIX 9

JUMP LOCATION IN A 0.15 m DIAMETER PIPE AT SLOPE 1/300,

CARRYING 6 £/s, FOR A RANGE OF MANNING COEFFICIENTS 0.009 to 0.018.

RESULTS PRESENTED FOR UNIFORM MANNING COEFFICIENTS IN BOTH APPROACH PIPE AND

TEST PIPE AND CONSTANT APPROACH PIPE MANNING COEFFICIENT OF 0.015.

A9.1
COMNJN

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A9.2
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A9.5

APPENDIX 10

SAMPLE OUTPUT PROGRAM HYDJUMP . THE OUTPUT CONTROL PARAMETER

SET TO GIVE WATER SURFACE PROFILES IN BOTH THE APPROACH PIPE AND

UPSTREAM AND DOWNSTREAM OF THE JUMP IN THE TEST PIPE

A10.1
PREDICTION OF ThE HYORAJLIC

JUMP POSITION IN A CIRCJLAR

CROSS SECTION PIPE AT HILO SLOPE.

PIPE LENGTH * 2.0000 H

•

MANNl >13 COEFF . --- 0.0120

FLOWRATE = O.OOfcO H$*3/S.

NORMAL JEPTrt * 0.0231 M.

CONTROL IS UPSTREAM, DEPTH =

PIPE WIDTH OR DIAMETER = PIPE SLOPE = 0.5000

CUNTkOL DEPTH = 0.0707 M.

CRITICAL DEPTH = 0.0707 n

0.0707 rt.

STANCE DEPTH ENERGY F+H

0. 0.0707 0. 0980 6.7950

O.OOOi 0.0691 0.0981 6. 8006

©.DOOo 0.0675 0. 0983 6.8173

O.OOIH 0*0660 0.0987 6.8663

0.0325 0 • 06 66 0. 0993 6.8681

0.QQ62 0 . 06 2 e 0. 1001 6.96 36

0®0063 0.0612 0. 1011 7.0135

0.0030 0.0596 0. 1026 7.0990

0®0I23 0.0581 0. 1061 7.2010

0.0163 0.0565 0 « 1060 7.3209

0.02 L2 0 . 05 69 0. 1086 7.6601

0.0271 0.0533 0.1113 7.6201

0.0362 0.0517 0. 1166 7.0029

©.0625 0.0501 0. 1 166 8.0106

0.0525 0.0686 0. 1236 8.2652

0® 0663 0.0670 0. 1289 0.5100.

0.0733 0.0656 0. 1355 8.8079

0.0950 0.06 38 0.1633 9.1628

0.1150 0.0-422 0. 1526 9.5191

0.1390 0.06 06 0.1633 9.96 1

8

0.1630 0.0391 0. 1 762 10.6171

0.2033 0.0375 0.1916 10.9526

0.2669 0.0359 0.2100 11.5562

0.3012 0.0363 0. 2321 12.2393

0.3702 O.C327 0.2590 13.0166

0.6600 0.0311 0. 2917 13.8981

0.5307 0.0<96 0. 3321 16.9096

0. 7516 0.0280 0. 3823 16.0736

1.0150 0.C2 66 0.6653 17.6226

1.5032 0.02 6e 0. 525b 18.9969

2.0000 0.C266 0. 5601 19.2662

0.1500 M

.

A10.2
PIPE LENGTH = 50.0000 PI. Banning cqfff. *= 0.0120

FL0WR4T:: = 0. OOP-O ".**3/S.

PIPE WIDTH 3* OlAflETER » 0.15 j 0 fl

.

PIPE SLOPE = 0.003 3

CONTROL DEPTH = 0. 0257 M

«

NORKAL DEPTH = 0.0 865 li

•

CONTROL IS UPSTkEAP, DEPTH =

STANCE CEPTH ENERGY

0 . 0.0257 0.5286

0.5772 0.0278 0. 3879

0.9597 C.0309 0. 2978

1.5555 C .C 3 39 0. 237o

1.923? 0.0370 0 • 1966

2.5095 0.05C1 0. 1676

2.3331 0.0531 0.1570

3.3559 0.0562 0. 1320

3.7935 0.0593 0. 1211

5.2212 0.C523 0.1133

5.6222 0.0555 0. 1076

5.9333 C.C565 0. 1036

5.3102 0 . Ot 1

5

0. 1009

5.5722 0.C656 0. 0992

5.7533 0 • 0t>7 7 0. 0983

5.8237 0.0707 0. 0980

CRITICAL DEPTH = 0.0707 «.

0.0257 fl.

F M

19.0558

16.1981

15.0535

12.50 72

11.1211

10.1053

9.2958

8.6501

6.1359

7.7295

7.5131

7.1731

6.9983

6.8620

6.8158

6.7950

A10. 3
PIPE LENGTH * 50.0000 rt. PIPE WIOTH OR DIAMETER - 0* 15oO «.

MANNING COEFF. * 0.0120 PIPE SLOPE » 0.0033

FLOWRATE - O.OOfcG N**3/$. CONTROL jE PTH - 0.0707 M.

NORMAL DEPTm » 0.086b M. CRITICAL DEPTH - 0.0707 H.

CONTROL IS DOWNSTREAM* JEPTrt * 0.0707 M.

DISTANCE DEPTH ENERGY F*M

0. 0.0707 0. 0980 6.7950

0.0105 0.0718 0.0981 6.7979

0.0529 0.0728 0.0982 6.8097

0.1015 0.0739 0. 0983 6.8169

0. 190b 0.0750 0.0985 6.8326

0.3169 0 • C 7 1

C

0. G9n7 6. 8521

0.5330 0.0771 0.0990 6.8768

0.7156 C o C 7 c 1 0. 09Q3 6.9056

1.0120 0.0792 0.0997 6.9383

1® A 0 2 3 0.0603 0. 1001 6 . 9 7 A 9

1.9202 0.0813 0. 1006 7.0151

2.6237 0 . Cfc 2 A 0 • 1009 7.0591

3.622 7 C . C b 3 4 0. 1015 7.1066

5.1556 C.Ct 65 0. 1019 7. 15 76

6. 0673 0.0856 0. 1025 7.2119

0.0000 0.0666 0. 1030 7.2696

JUMP POSITION

CONJUGATE DEPTHS

KINETIC ENERGY VALUE

HYDRO ST A TIC* MCMt NTUH

$.8557 M FROM PIPE ENTRY.

0®0573 M UPSTREAM AND 0.0bb5rt DOWNSTREAM.

0. 1051M UPSTREAM and 0.1029 n DOWNSTREAM.

7.2612 N.

DEPTH Ci AN Gt AT JUMP = 0®2911bE“01 M.

ENERGY LOSS AT JUMP * -©.21271E-02 M.

A10. 4
APPENDIX 11

PROGRAM HYDJUMP

All . 1
PROGRAM HYDJUMP

A full print out of HYDJUMP is included in this appendix. The program was written in Fortran for use on the NBS CBT Perkin Elmer 732 computer. No special

facilities are required, single precision sufficient.

The program contains numerous comment blocks describing the calculation technique, however a simple flow diagram is included in this appendix.

Input data to the program is simple, the program being designed to carry out numerous repeat passes to generate the tabular data included in earlier

appendices. In order to facilitate use, a set of example data is included in

this appendix for a number of options.

It should be noted that all calculations are carried out in SI units. All

input data are in SI units with the exception of flow rate; this is entered in

litres/s and converted within the program to m-^/s.

All.

2
H YsiUrA Y

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4
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oIOAI..1

PROGRAM HYCJOMP PtFERS TO A Pf- LG PA*4 OcSIGNEO TU

CALCULATE THE HYDftA-JLiC JUMP POSITION AFTER A CHANGE IN

SLOPE IN EITHER A R-CTANGUL/K GR CIRCULAR CROSS

SECTION PIPE.

DIMENSION >(2,100)»F(2,100),PP(2,2),M(G)

DIMENSION CP(2,2),ER(2, 100) ,Dt P ( 2 , 100 ) , I P ( 2 ) ,rtJ ( 2 ) , E J( 2 I 01 ME NS I ON 07 ( 3 GC ) , S 1 ( i 00 ) ,PN 1 ( 3CC ) , S2 ( 300) ,HC2( 300)

0 1 ME NS ION HN213C0), T EN 1 ( 3uC ) ,£EN(300 ) , HE N ( 300) , HUP ( 30 J ) DIMENSION PCM 300 ) , E J° (30u),ELN(3G0),wFPM( 300)

01 ME NS ION FFM300) ,3 JH( 300) ,Cl J (3L0 ) ,X JUMP ( 330)

INTEGER SmAFE

comm ON /Cm:/ P ,C,G,C3N,S 3,GAM,FhG,HCR IT,HNOkM, AkE A .Pc R ,FPm,ENERO

C0MM0N/CM2/SHAPE

COMMON/Ch 3/ I Z READIA ,702 ) IJCUT

RE AD ( A ,702 ) SKA PE

C IGuT ISA OUTPUT P 3 R 1 A T CuMP.LL C0D£,=1 OnLY uUTPUT

: IS A Sli^ArY 7 A I L c » = J OUTPUT INCLUDES RATck SU^FAUE

C PROFILES.

702 FORMAT ( 13)

902 READ (A , 120)NN,NS,1C3 I RE A D C A , 1 2 1 ) E , K P C , R.1 1 » ) " N

121 roRM A T ( A F 1C • <• ) .1=9.81

\0cC

n

v*

v#c

wr

c IC*0

I COUNT = 0

IZ = 0

300 CONTINUE

IF ( I Z. EQ. 3) GUT C 9013

9011 CONTINUE

I F ( I7.CT.1) IMIZ)«I S IZ = IZ> 1 IF( IZ.EO. 1) Wfi ITE(3, 50A

)

50 A FORMAT ( 1H1, /////)

I Ft IZ.EQ.l.*N0.SHA D E.£3.i ) H F I TE ( 3 ,5 05

)

505 FORMAT (//////////, 2 OX*'PKEf ICTICN OF Tut HYDRAULIC',

l/,20X,'JUMF POSITION IN A RECTANGULAR ' , / 2 G X , 2 'CROSS SECTION PIPE AT MILD SLCFF.' ,///////)

IF (SHAPE. EC .2. A NO. I Z.SO.DViFITE (2,500)

5 00 F JR MAT ( 20 X, • F s EC I OT I'M OF THE HYCk AUL i C

* ,/ » 20a

,

l * JUMP POSITION IN A Cl .RCULAR ,2CX , 2 ' CROSS SECTION PIP 1 ' AT M I Li. SLOPE.',/////) *

I F ( IZ.GT.3 ) 17=1

IF(IZ.EG.i) 1C= I C

1

reao(a,iog)pl,so,c,h';jnt,enz
wC IF ICON-2 THF ANALYSIS IS PAStD ON THE ASSUMPTION OF

C TERMINAL FlTW CONDITIONS AT THE DISCHARGE FROM THE

C APPROACH PIPE.

C ICON = 2 ALSO INTRODUCES THE CONSTRAINT THAT TrtE

C TEST PIPE IS LONG ENOUGH FOF NOFMAL FlOH DEPTH

C TO BE ESTABLISHED DOWNSTREAM OF THE JJMP . NORMAL

c oepth then ffcomes the downstream conjugate depth

C WHICH allows calculation of the •f + m* term without

c calculating the water surface profile downstream uf the

C JUMP.

C IF I CON = 1 THE FLOW ANALYSIS IS PASEO UN AN INPUT ENERGY AT

C ENTRY TO THE MILD SLOPE PIPE, TEFM ENZ . SIMILARLY THE lOSS

C COEFFICIENT AT PIPE ENTRY MAY b F lX.PRESSEu AS A FaCIOkiDENi

C VALUE 0 TO I.O, TO OE MULTIPLIED BY THE ENZ TERM.

C AS the entry flow depth is tc bf calculate 0 FROM ENZCjEN

c and nut based cn flow critical depth, the control term

C HCONT IS SET TO 1.0. THE uT HER PARAMETERS REFEh TO PIPE Fl'JW

C RATE AND DIMENSIONS.

100 FURrtAT (5F1G.4)

IF { PL.EO.O.C JCOTG 0 JI

C PL”PIPt LENGTH, t —W I > T H , RM-MANMING C'JEFF, SQ-SLOPE,

C J-FLCW RATE IN L/S TO 3 E USE T IN M**3/S, HC ONT-CONT A OL DEPTH

C Sc T ID ZE*l If CRITICAL DEPTH ASSUMED.

0*Q/1000.0

•i '1 0 = 1 0 O O . (

j

IF(IZ.EG.l) F r, = Rnu

I F ( IZ.GT.i! Fm«FM1

C0N=Rm$*2/SC

GAM-G A RHO

W

r

w

c JTC ICi -QS1GCC.0

IFUZ.E0.DS1! IC J-S3

IFUZ.EQ.2IS2C I C J = S 0

C PROGRAM CONTROL DATA.

' 120 FORMAT C 3 I 6 ) C NN-SIZE OF THE CH STEP IN SIMPSCNS RULE,NS~NO.

C STEPS IN DFPTh CAl‘C.,ICON DFTtPMNES WHETmER EnERoY

C INPUT OP UPSTPE/M SLOP? IS USLC, L F N IS THE ENERGY

C LOSS FACTOF PDF ThE PIPE ENTRY, IN EIT.iER cnTRY CASE.

C ENZ — IN PUT ENERGY TO REPLACE SLOPE If ICON = I.

IF CIC0N.EC.1I GOTO 650

GOTO 651

650 £NZ=DEN«ENZ

CALL BOUND I ENZ , 0 , B , H 3 I

I Z = 2

HCONT=HB

GOTO 18

651 CONTINUE

IF (HCONT.EC. 1.0 I GOTO 17

GOTO 18

17 ENGD-=D£N$cNGD

CALL aOUNO(FNGD,U,B,HB ) HCONT-HB

18 CONTINUE

C. All. 10
c

c

cc

C

" cc

C OETERMlNATl CN OF CRITICAL ANC NCTHAL DEPTHS.

C THIS SECTION CALCULATES THE NO R HAL AND CRITICAL OE PTH IN

C EACH PI 9 £ LFMGTh FOR LATER CCnPAH.SDN TO THE CONTROL jEPTh

C INPUT.

C CALCULATION Cr CRITICAL DEPTH.

UP =P

i)N = 0.0

HC=UP/2.0

7 CONTINUE

CALL CALC(HCtLL)

IF1HCR I7)3,4,i>

3 J.i = HC

GOTO t» > :jp=hC

b .iCN={JP*jnJ/Z«C

IF (A3$ ( (HCN-HC I/HC) .LE .0.001) OCTl 4

= 'J C s

vi JTu 7

3 HC’HCn

A IF(HCONT.EC.C.O)HC )NT = HC

C CALCULATION cf normal depth.

UP = 3

UN =0.0

HN=UP/ 2.0

9 CONTINUE

CALL CALC(HN,LL)

c

cc

c

c

f+ s wcc

c IF(HNORM) 10,11*12

10 ON=HN

GOTO 13

12 UP =H N

13 HNN=(UP*DN) / 2.0

IF (A3S ( (HNN-HN )/HN) .LE .O.UCl) CCTo 14

1N = HNN

GOTO 9

14 -4N = HNN

11 CONTINUE

IF ( IOJT .EC. 1) GCTo il ) WRITE! 3 , 20G ) PL , f , K M ; i>)iMCIFT

All. 11
200 FORMAT ( ////2CX , *P IP6 LENGTH - a «FlG.4« a H.*t5Xt

l

a P I PE WIDTH CR DIAMETER * , ,F1G.9,» M. * , / , 2 OX ,

• MANN I NG COE F F«

2F1Q.9,5X,*MFE SLU»S * « if 1 0 . A , / , 2 LX , • b LG W* A T t =*

3F10.9,* M*«3/S • N3X. 'CONTROL DEPTH **,F10.9f* H.*,//)

WRITEC 3,201 >FN,HC

201 FORMAT C20a,*NGPMAL DEPTH **,F10.9,* M . • , 9X ,

• CR I T I C AL DEPTH *•»

IF 10 • 9 *

* M .*,//)

C FORMATION CF A HYDRA JL IC JUMP RFCUIRES THAT DEPTH

C CHANGES FROM btlGw CRITICAL TO ABOVE CRITICAL* HENCE

C IF NORMAL CFPTH IS LESS Than CRITICAL NO JUMPFGRMS.

IF ( I Z .GT. l.AND .HN.LE.MC ) WRI TE ( 3t?L9)

209 FORMAT C / * 2 C X *

9 JUMP FORMATION IMPOSSIS-E AS HNCHC.*,/)

310 CONTINUE

C

rW

r

wcc

c

c

**> c

' •

IF( 1Z.EQ.1IHNK IC)=HN

IF C I Z» GT. 1 )HN2 ( ICJ *HN

IF< IZ.GT.1)HC2( IC)=HC

IF(IZ.GT.1.AND.HN.LE.mC)GQTC eoc

C FULL BORE FLOW IS NOT DEALT WITH cY THIS PROGRAM.

C If THE VALLE CF HN > PIPE DIAMETER SOLUTION IS

C TERMINATED.

IFUZ. hO. j. A Nu . ICC 9.E J . 2 > CCTL 9 Ac

GOTO 997

C IF THE TEST PIPE IS LONG THEN A CLNSIJEPAdLE S IM P L I F IC AT I D N IS

C POSSIBLF AS THE FLCH DEPTH MAY PE ASSUMEO TO BE •NORMAL*

C DEPTH ANO THE ( F M) VALUE AT ThE JUMP IS FIXED.

996 CALL CALCCHN»DLI

F X - F PM

EX2 =EMc R G

HX 2 = HN

HL I M*B$0.99

IF CHN.GE.hLlMl HN2(IC>=3

I F C HN. GE .HL ! MJ GOTO 300

DO 999 J 6= 1 , IS-1

C IF THE F M TERM AT ENTRY IS LESS THAN THE F+M VALUE

C FOR THE NCFMAL FLOW DEPTH THIS INDICATES THAT THE

C JUMP HAS NCVED TO L»9 AS A CkLKNED JjrtP

.

FEN ( IC ) =F ( 1 , 1)

IFtFX.GT.Fflii) i GOTO 330

IFCF(1 »J6).CF.FX.AND.F (l*J6+l)«LF *FX) GOTO 99d

999 CONTINUE

C

ccc

c All. 12
HXi*DE P(l,JM*R«<UE t> (l,J6*l)-0EP(l,Jb))

EXl»EN(l,Jb)+R«(EN(l,J6*l)-fN(l,J6))

CHE-EX2-EX1

IF(IGUT.EQ.l) GOTO 311

WRITE! 3*952)

952 FORMAT ( /20> , ’TEST PIPE ASSUMED LONG* NORMAL FLOW OfcPTH *,

l *E STAB L I SHE C DOWNSTREAM OF TFE JUMP.*,/)

WRITE (3,107)XX,HXl,rt<> *E X 1 , F X2 * FX ,CHH, CHE

GOTO bOO

311 CONTINUE

HUP ( IC ) =HX

1

HON C IC )*HN

£ UP (IC )=f xl

£ON( IC ) *£ X2

OJH(IC)-CHH

0€ J( IC )=CHE

CFPM ( I C ) = FX

X JUrtP( I C ) = X X GOT 0 -iOO

3 33 IF ( I OUT. EG. C) WRITE ( 3, 3 32 ) 332 FORMAT </2CX, 'JUMP DROWNED AT L*C’,/)

Ft S ( IC )=F (1 ,1)

CF PM ( l C ) =F

X

GOTO BOO

C

C

C

f*wC

c

rC

957 CONTINUE

C THE APPROACH PIPE LENGTH MAY bE IGNORED IF TERMINAL

C CONDITIONS AFE ASSUMED. TriIS SECTION jSES THIS OPTION

C 3 Y CHECKING ON THE ® I P E NUMBER ,12* AND THE VALUE 0

F

C ICON WHICH IS SET TO 2

IF( IZ.EC. l.ANC. ICUN.E9.2) GCTu 95 3

GOTO 955

953 H=HN

CALL CALC(H*DL

)

ENGO=E NERG

I F C IOUT.EQ. 1 )GCTO 312

WRITE! 3,95* )H, ENGD

955 FORMAT ( /20X* 'TERMINAL VELOCITY CCNDITlUNS IN APPROACH PIPE.',

1/20X, •DEPTH * • ,f ID • 5 *

' N

.

' , 5X ,

' F NE P G Y = ' *F LO • 5,

•

GOTO dOO

312 CO NT I SUE

TEN1 ( I C ) =E NGC

GOTO 300

955 CONTINUE

Hl IM=0.99$B

IF ( IOUT.FO. 1) GOTO 313

IF (IZ.GT.l. AND. HN. G G.HL !*> WF ITf (3,790)

313 IF (HN. GE .ML IH) HN2(IC)=A

IF ( IZ.GT . 1. AND .HN.GG.ML I.M )GCTL PCG

All. 13
Lc

c

ccC THIS SECTION PREPARES FOP THE PROFILE CALCULATIONS

C iY SORTING FLCWS BA Sc J ON PIPE SLOPE ANO THE NORMAL AnD

C CRITICAL DEPTH VALUES.

IF (HN.LT.nC ) GOTO 50

C MILO SLOPE.

IFIHCiJNT.LT.HC I GOTO 45

C SUBCRITICAL FLOW, MCQNT GT. HC

.

S I GN*- 1 .

0

DH* (HCONT-HH )=»G.99-5/FL0AT t NN)

IF(IOUT.tO.C) 'WRITE! 5,202) HCCN

T

202 FORMAT !20X, 'CCNTRGl IS DOWNSTREAM, DEPTH *',F10.4,' M. • I GOTO 60

C SUPERCRITICAL FLCw, ^CONT LT. HC

.

45 SIGN =1.0

JH=(HC-HCuNT ) / F LCAT (NT ) IF ( I OUT . t 0 • f ) WRITE! 3, 203 ) I CENT

203 FORMAT (20X, 'CONTROL IS UPSTREAM, L

r P T H * # ,FI0.4,' M .

'

)

GOTO 60

C STEEP SLOP E , H N LT HC.

5 C IF (HCOnT.LE .HC ) GOT 3 55

•: sjjCri tical new, hcqnt gt hc.

S I G N = - 1 . C

OH*!HC uNT-FC )/FLCAT(NN ) IF( IOuT.Eu.O) WRITE ( 3,202)HCCNT

GOTO go

cc

c

r

v*rcc

c 9 c

cc

C SUPERCRITICAL FLOW, HCCNT LT r- C.

35 SIGN=I.C

N2*NN*2

0H=(HN-HC0NT)«0.995/Fl0AT(N2

)

IF { IOUT.EQ.C) WRITE! 3, 2Q3JH.CGM

60 SL- 0 .

0

I S* 1 H*HCON T CALL CALC! H.,LL I t * E NE R G

fm=fp

m

IF C IZ.GT. 1 )F (I Z— I , I S ) * F

1

IF(IZ.EQ.2)>( 1,15)0.0

IF ( IZ. CO. 3 )X U , IS )*!>L

IF( IZ.GT. I)DEP ( IZ-L , IS ) *H

IF ( IZ.GT, 1 JEN! IZ-I, 131 *S

All. 14
oooooooooooooo

IF € IOJT.eO.l) CCTO il<t

W*ITE(3,2G4)

204 F Q R fi A T ( / / / / 1 19 X *

' DISTANCE CEPTI1 'i' ENERGY # t* FM'I

rf*lTE( 3,205)SL,h,E #Fi

20^ FoRHAK ldX,4F 10.4)

314 IFC IZ.E0.2IHENC ICi»H

1FC IZ.EQ.2 >EEN< IL)*S

1A TER SUFFICE ERlflLl CALCULATION 031*0 S I P 3 ur; S

TO EVALUATE TFt INTEGRAL.

33 00 1 = 1, I S ,2

SLO=SL

IS*IS*1

ri2-HCCNT»S ICN*DH*FL3AT ( I 1

1

-» 3 »HC j NT* 5 1 COFLC AT (I J *3n

CALL CALC(F,LL)

CALL CALC(h2,LL21

CALL CALC(H3,LL3)

OX»DHS <DL*Cl 2*4.0*0L3) / 3.G

SL*SL*CY

IF (SL. GF.PL )GCT 0 b2

i1=H2

CALL C‘ALC(F,DL)

5 =E NERG

F M = FPrt

904 CONTINUE

IF ( IZ. t 0. 1 . ANL .H.LE . M >G07C 911

GOTO 912

911 Si_ * PL

:t=HN

CALL CALC(H,DL

)

E = E NEk G

F M = FPH

912 CONTINUE

IF < IZ.EQ. 3. AND.h.GE .3) GOTO fill

HHX=AdS ( (H-FS) /HM

IF ( IZ. EC. 3, ANX-HHX.L5. 0.001 JSL=PL

dll CONTINUE

c

r

wc

c

cc

c

n * w

All. 15
IFCIZ.GT.1)FCIZ-1,ISHFM

IFUZ.EQ.2)Xi,IS)*SL

IFC IZ.E0.3)>(2* ISHPL-SL

IFC I Z. GT • l ) TE P C I Z — 1 « IS ) *H

IPi I Z • CT • 1 ) E M C IZ-1» IS) *E

IFCIZ.fQ.2./NC.h.GT.HC) GOTO fcOC

IFC IZ.EQ.J.ANU.H.GE.-H GOTO BOB

IF (IOUT.EQ. 1) GGTU 315

WRITE(3,2C5)SL*H,EtFi

315 CONTINUE

iFdz.Eo.i. anc.sl.ge.pl ) gotl eoi

IFC IZ.EO.I.ANB.SL.LE.SlO) gctl t Cl

IF CIZ. EQ. 3. /ND.SL.G?. PL) GOTO 90C

SO CONTINUE

SOI ENCODE

IFCIZ.fQ.3) Co T C 90 0

GOTO 800

32 H s rt2 — S IGN^.G^LH^CSL-PL )/0X

CALL CALC(h,LL)

£

3 E N £ R G

F rt s F Pn

E n G 0 = E Su s PL

GOTO 9C

A

30b CONTINUE

C

rCC

C

C THIS SECTION CEALS WITH THE POSSIBILITY OF FULL SORE FLOW

C St CONING ESTABLISHES IN THE PIPE GOWNS TRE Art. OF Tr.t JUMP

C LOCATION. THE POSITION OF THE HYCMUlIC JUMP IS

C DETERMINED PY EGUIvalSnCE CE THE TEPrt BETWEEN THE

C FULL 8 CRE FLOW AND HE UPSTPfAH SUPERCRITICAL FLU*.

IFC SHAPE. EC. 1 J AREA-3**2

IFCSHAPE.EC.2I AREA = C 3. 142*E**2 )M. 0

IF ( IOUT.EQ. 1) GOTO 3la

WRITE! 3,790 I 790 FORMAT J/,2CX, ’FULL 3 JR £ FLOW E ST Ac L I Srt E 0

.

* /

)

316 CONTINUE

ENC2 * I S )*3*(C$ *21/ ( p.OSG^AP.E A**2.)

FC2, IS )=RHO*C*C/ARE A^-GArtPARE A«e/2.0

IF (IOUT.EQ. 1) GGTG 317

WRITEC 3,2C5 )SL,B,EN(2, IS ) ,F(2,IS)

317 CONTINUE

XL “PL— SL

i S * I S L DEP (2, IS HE

X ( 2 « I S HC.G

SL-PL

£N ( 2 * I S ) *E N i2 $ I S— I

D

F(2f IS )»RhC* QPO/AREA^G AM* AP.tAM E/2.0)

IFC IOUT.EQ. 1) GOTO 3171

WRI TEC 3,205 )SL9B,EN(2t IS) ,F(2,IS)

3171 CONTINUE

GOTO 900

900 CONTINUE

C

All. 16
oo

oo

o

IDENTIFICATION CF JUMP POSITION eY EQUIVALENCE

•JF THE F m TERN.

IFIFCI • I) .LT.F ( 2 * IS ) • \NO . 1 OUT .E O.O) WRITE ( i • I 385 FORMAT </20X, 'JUMP JRQHNED AT L *0 ATTEST PIPE fcNTkY*,/)

IF(F (1*1). LT.F (2, IS > ) GOTO 8C0

XP*0.0

0ELX*PL/50C.0

103 XP *XP + DELX

00 102 1*1*100

J*2

IF <X{ J , I 1 >.LE .XP.ANJ. X( J, I ) .Cfc .XP I GOTO 108

IF(X< J* 1*1 ) .EC. 0.0) GOTO 103

102 CONTINUE

105 CALL INTER (X( J , I 1 ) .<=( J* l + l ) ,X ( J, I ) ,F < J, I ) *XP,FX)

J*1

00 105 K* 1,1 00

1FIX.GT.Im

(

2)i GOT J 103

IF(X( J ,K ) .Lf .X P.AnO. <( J*<* 1 ) .cT.XPl GOTO 109

GOTO 105

109 CONTINUE

IF(F(J,K).CE.F X.ANO .F l J , <. 1 ) . L t . F >. ) GC T ; 110

105 CONTINUE

C

cu

ccc

110 CONTINUE

FP(1,1) =F( 1 * K

)

FP (1*2 )=F ( 1 ,K>1

)

OP (1*1)=X(1*K)

OP (1*2 )*X( 1 *K*1 ) FP(2*1)=F(2*I-*1)

FP(2,2)=F(2*I)

0P(2*l)=X(2*l*l)

0P(2,2)=X(2*I>

I P ( 1 ) = K

I P C 2 ) = I CALL SOLVE'. 0P(1,1)»F?( 1*1)*DP(1*2),FP(1,2),0P(2,1),FP(2,1) * 10P(2*2i *FP(2,2) ,XJ,f= J)

C CALCULATION CF DEPTH AND ENEPCY CHANGES AT THE JUMP.

DO 59 K = 1 *2

CALL INTER(X(K*IPCX)),0EP(K,IP(K)),X(<*IPIK)»1),

IDEPtK, I P l K. ) 1 } ,XJ*HJ(<) ) CALL lNTEk(X(/,IP(<n,EN(K*IP(K)),^(!<*IP(<<)«-I),

1EN(K,IPCK)*1),XJ,EJ(XI)

59 CONTINUE

C

C

C

C

All. 17
oooooooooooo

CHH-HJ 4 2 J — H J(1

J

CHE»EJ(2)-E J< 1)

IF ( [GUT .E(J. 1 } GOTO 3U

HRITE<3,L07)XJ,HJ1L),HJ<2),EJ( 1),£J(2) « FX,CHH,CHE

107 FORMAT (////, 20X,*JJ1P POSITILN ••Fl0.-»»

i* M FROM P IPf ENTRY. 9 , 1/ » 20 X, ’CONJUGATE DEPTHS - *,F3.4, 9 M UPSTREAM ANJ •,

IF 6 • A *

* M DOWNSTREAM. 9 ,/, 20X,

I*KINETIC ENERGY VAL'Ji = # ,F8.A, 9 M UPSTREAM AMD *,F9.<»,

i® M DQWNSTRF AM .

9 ,/ , 20X

,

l ®HYDRUSTA T I C MOMENTUM » 9 ,F9.<,, 9 N. 9 ,/,

1/ , 2 OX, 9 PE P Th CHANGE AT JUMP = , ,E1*.5, 9 *.*,

2/,20X, * ENCRCY LOSS \T JUMP * 9 ,E 1*.5» 9 M. 9

I 313 CONTINUE

HUP C IC I =HJ ( 1 ) HON I IC I -H J ( 21

EUP IIC ) =E J I 1 »

’ EONCIC )=E J f 2 i UJti( IC i =ChM

0£ J< IC )=Cr!fc

CFPiltI C i =FX

X JUrtPC IC )=X

J

IZ = 0

GOTO 400

9010 CONTINUE

IF 4 IOUT.EU. 0) GOTO 320

C THIS OUTPUT SECTION IS ONLY USED IF I UUT=1 , T A3 UL AR

C OUTPUT ONLY

IF ( ICON. EU. 2. A ND.ICOUNT.EQ.O J WP!TEC3,325i

325 FORMAT! *1 PPCGRAM RESULTS BASED CN TERMINAL CONDITIONS 9 , 1* IN THE APPROACH PIPE AND NCkM/L FLOW DEPTH JU HNS T R E AM 9 , 2® OF THE JUMP IN THE TEST PIPE.*,//)

IFC ICON.EQ. 2.AND.ICOUNT.EG.O) WRITE (3,321)

I * I C

IF C ICOUNT. EC. 2 0 . AND . IC0N.EG.2)HPITE(3, 32 5>

IF ( I C0UNT.EC.2OJWR I TE( 3,321

)

IF(IC0UNT.EC.2G) I COUNT =0

ICOUNT= I C 0 U N T

1

IF CHN2 ( I ) .LT.HC21 I ) ) SOTO 370

IF (HN2 (I ) .E C.u ) GOTO 371

IF (FEN < I ) .Lr .CFFM.(l) ) GOTO 37^

WRI TE( 3,323 )GT (I) ,3 ,Ri, SI ( I ) ,hNl( i) ,TcNl( I ) ,S2( I ),HC2(I)

,

. 1 HN 2(1) ,HE.>t(I),EF. N(I)|FEN( I), HUP (I ) , HD.MI I ) , 0 JHI I),EUP(I),SDN(I>,

2JEJU) , CFPM ( I) ,X JUW «M l ) GOTO 322

3 70 RRIT6C 3,330 )QT ( I ) ,3 ,P1,S1( I ),HN1 ( I ) ,T£N1( I ) ,S2 ( I ),HC2( I ) ,H32 (I)

All. 18
371

372

322

r

u323

330

381

382

321

320

101

GOTO 322

RRITE( 3.381 )CT< I ) ,3 .Ri,Sl( I )*HNl ( I)

,

GOT 0 322

RRITE( 3,3d2 >CT( I >.B ,R1.S1(

I

>.HNl( II . GOTO 322

CONTINUE

TENICXI tS2C! )tHC2CI >»HN2U >

TcNl(X) *S2( I )*HC2(I ) . MN2 ( I

)

F0RMAT(/*X*F4.1'F5.2«F6.3»F7«'t'2F6.3tF7.4»3F6.3tF6.2t3F6.3«

1*F7. 3.2F6.3)

FORHAK /5X.FA. 1.F5. ’'Fb.3,F7«4'2FL. 3*.- 7.4.2Fb.3*15X«

i» JUMP IMPOSSIBLE AS HN<HC IN TEST PIPE.*)

F OR" A T ( /SX.F A. I,F 5. 2,F*. 3.F7.A .?Fb. 3.F 7. A, 2F 6.3, 15X.

1* FULL B0.«£ FLOP EST A 3 L I S HF D IN TEST PIPE.*)

r JRMAT (/5X.fA.ltft) # 2»r5.3tF7» <**2F6. J.^T.^.ZFb.i.I^X.

l* JUMP DRUPNED AT L = 0 AT TEST PIPE ENTRY.')

FORMAT < ///SX, • COM 1 0N DATA ', 'APPROACH PIPE DATA ',

125X.*TEST PIPE DATA AND PkCGRAM P E. SUL T S. ' » / / . 23X*' J. OlA. MANN. S'. j°E FN TE°“. SLOPE rC HN

3'ENTRY ENTRY ENTRY 'JPJ'JMP clpn depth fnergy energy en^gy

<»* JUMP JUMP './.DX.'L/S M • CftFF iSlNi n . cNERGY',

3* (SIN) N . h. DEPTH ENtKGY f N DEPTH ‘JtPTh CHANGE • b'UPJUMP DOWN CHANGE F . PuS. './,5X,

72d X *

' M. ' , 19X. ' M. M. N. ft. ft . M. * » 3 ' ft. M • *1 • N. M. • » /

)

C G N T I N U F ,010

CONTINUE

RE AD( < , 702 ) Sr-APE

IF ( SHAPE ,GT .0) GOTO 102

ENO

»

C

SUBROUTINE INTER (A» 3,:»0»X,Y)

C SUBROUTINE INTER SIMPLY INTERPOLATES LINEARLY bETWEEN

C TWO SETS OF DATA POINTS ANC IS USED TO CALCULATE

C JUMP DEPTH AND ENERGY CHANGES AS PELL AS PUSITlON.

Y *& ( X- A ) $ ( D-b )/(L-A)

RETURN

END

C

C

SUBROUTINE SOL VE(X1*YL , X2 • Y2 .X 3 . Y3 . 'H* YA . X 5 . YS

)

C SUBROUTINE SOLVE DETERMINES ThE INTERSECTION POINT OF

C TRu STRAIGHT LINES DRAWN i F TPfc E N FPM-X COORDINATES

C IDENTIFIED AS LYING ON EITHTR SICt OF ThE JUMP POSITION.

A- ( Yl- Y2 ) / ( > 1-X2

)

3 * Y l — X 1 *A

C * ( Y 3 — Y *» ) / (X3-XA)

J = Y 3 — X 2*

C

<5* ( D-B ) / ( A— C)

YS^AOXS^B

RETURN

END

c

Cc All. 19
c

c SU3RQUTINE p CU Nli ( t , Q , 3 * H B

)

INTEGER SHAPE

CUHM0N/Cr2/SHAPE

C SUBROUTINE EOUNL CALCULATES ThE'ENTRY CONDITION TO THE

C HI LD SLOPE PIPE <IZ*>> BY REFERENCE Tu THE ENEkGY AT

C JISCHARGE FED* THfc STEEP SLOPE PIPE U2=l). OR SIptPLY

C FROM ThE ENERGY IN^UT DATA IF THAT MQjfc IS CHOUSEn

C BY THE INPUT OF ICON 1 I IN TrE INITIAL READ STATEMENTS*

C PIPE CROSS SECTION IS CONTROL LLO tY THE VALUE OF

C TERM SHAPE* 1* RECTANGULAR OF 2*CIfC'JLAK, iN TnE INPUT.

G 3 9. 81

IF C SHAPE. GT.l) GOTO 2

Y1 s O oQ

BB=0/£ ti*< 2. 0*G*£ >**0.5)

Ort 3 H3/2O0.C

HX=HB/2.0

00 75 1*1, ICO

IF(I.GT.l) Y1*Y

HX=HX*DH

Y=(0$*2J/(2.G«G)*((3* i< )«=*2 )*(HX-E)

IF (I .EG. 1 K ETu 75

IFIYI.Gc.O.C./NL.Y.Lw.O.JI GOTO 7 6

lF(Yl.Lf .C.C.AM.Y.Cf .0.0 CCIb 7t

75 CONTINUE

76 H3 =HX

GOTO 1 2 CONTINUE

rj= Q . 0

EC 1~0.

0

OELH=d/200 •

C

P£*3.LA2

R*6/2.0 x 3 H = H«-D£LH

IF IH.LT.RJ THETA* 2. 0*A TAN ( SORT (HM3-HJ ) /(R-HJ)

IFCH.EQ.R) THE TA = PI

IF CH.GT.R J THE TA = PI «-2 . 0 * A TA N ( ( H-F )/ ( S Ok Tin* (3-H J

i

)

)

AR£A=( ( P $ * 2 )/6.G)*(THcTA-SlN(Tf’FTA) > EC2 3 H«-(0**2)/UAFcA**2><‘2.C*CJ

IF(EC2.LE.E.AND.EC1.1E.E» CGTC A

EC1-EC2

GOTO 3

4 rlB*H

l CONTINUE

RE TURN

END

C

C

c

ccc

c

Cr All. 20
oooooooooo

ooo

l 3

SUBROUTINE C AL C ( H *DL ) SUBROUTINE CALC IS 'JSc 0 THROUGHOUT THE PkUGRAfl TO

JETERMI NE THE FLOW-PIPE PARAMETERS SUCn AS FLuW

OEPTH, AREA, WETTED PERIMETER AS WELL AS BEING USED

is the bisection met.hjd calculation of nokhal and

CRITICAL DEPTHS IN EACH OF THE PIPE LENGTHS.

In THE CIRCULAR PIPE CROSS SECTION CASE IT ALSU

CALCULATES SUBTENDED ANGLE AND THE .WATER SURFACE

WIDTH AS uEFTh CHANGES.

AS IN 80UNC AND MAIN PROGRAM THE PIPE SHAPE IS DETERMINED

BY THE VALOF OF ThE TERM SHAPE INPUT AS DATA.

INTEGER SHAPE

COMMON/CMl/P,Q,G,COS, S 0 , G AM , RhC , HCR I T , HNOR M , 4k fc A , P E R ,FPM,ENERG

COMMON/CM 2/ SHAPE

COMMON/CM3/ I Z IFISHAPE.GT.DGOTO I lF<IZ.£0.3.AND.h.CE.B)H=3

AR£A=H«3

P£R=B*2.0*h

NCR I T= I. 0- (C**2 )«B/ <0* ARE A**3)

HNURM= 1. 0-(C*$2 J *CJ W/((AR^A*«j.32j)/<p£K#«i.3ij))

DL = HCR IT/ ( HR Of* P*SC)

F ° -> = ( G A y * * F E / v F / 2 . 0 ) ( R ^ 1 v c * i /

:

t

t / ) f NESG=H> I «**2 ) / ( ( A*E A* *2 ) *>2 .c*G)

GOTO 2

p I - 3 . i s Z IF (IZ.^«f.j./Ni> .H.Gz.3) G J i i c C

IF IH.LT .R ) THE TA^2. 3*4 T AN (SORT ( H* |B-H J I / ( c/ 2. j-H) ) IF(H.Ew.R) THE TA = P

I

IF(H.CT.»J THE.TA-PI «?. 0*ATAM C f - F / ? .0) / f SuwTCn* < B- - > )) ) G J TO 2 2

20 1 = 3

THE TA= 2. 0«P I AREA = P I*<S/2.G)**2

PER=PI*8

X0*8/2.0

GOTO 21

22 CONTINUE

AREA=((B**2)/e.0)*{ THE T A-S I N ( T h E T A)

)

PER=8*THETA/2.0

T*2.0M <H*( E-H > )**•> .5)

HCR I T*1.0-(C**2.0)*r/{C*4RE£**3}

HNCRM= i.0-(C**2.0)SC )S/( ( Af< F A« « 3 . 33 3 ) / < PER **1.33 J) ) DL = HCRIT/(HNf. KM*SO)

X0 5*(2.G/3.G)«(b/2.0)‘>(3.0<: SlN(THETA/2.0)— SlN(j.0^TH p TA/2.0))

1 / I*. 0* ( THE T A/2.0-0. 5 *S IN (THETA) ) ) 21 H3AR=X0*H— t /2.

U

FPM=GAM* Akl /4HBAP.* kH 3* 0*0/ A PEA

ENERG*HMG*«2)/((A*EA4»2)*2.0»G)

2 CONTINUE

RETURN

End

i BEND

All. 21

APPENDIX 12

PROGRAM HYDSUM
PROGRAM HYDSUM

A full printout of program HYDSUM is included in this appendix. The program

was written in Fortran for use on the NBS CBT Perkin Elmer 732 computer. No

special facilities are required, single precision sufficient.

The program is designed to calculate the critical and normal flow depths based

on pipe slope, cross sectional size and shape, flow rate and Manning coefficient. No flow chart is included as the program is directly copied from the

early sections of HYDJUMP.

The output is in tabular form and presents critical and normal depths as functions of pipe slope and flow rate for set values of Manning coefficients.

In addition (F & M) values based on normal flow depths are calculated and

tabulated as functions of pipe slope and flow rate.

Sample data are included in this appendix.

All calculations are carried out in SI units and all data are input in these

units with the exception of flow rate which is input in litres/s and converted

in the program to m-Vs.

A12.2
Sample DATA Program HYDSUM

Case 1 Tabulation of hn , hc , (F+M) for a range of 8 flow rates for circular

and rectangular channels.

Line 1 SHAPE 1 = rectangular

2 = circular

Format 13

VV2

Line 2 T (diameter or width), RM(Manning coefficient), QMIN(lowest flow

rate), DQ(flow increment) NI(N° of test pipe slopes), N2(N° of

approach pipe slopes).

Format 4F10.4, 213

VVVVO. 1500 VVV VO. 0150VVVVl .OOOOVVVVl .0000 VV2VV3

Line 3 SI test pipe slope, Format F10.4.

VVVVO. 0025

Line 4 SI test pipe slope, Format F10.4

VVVVO. 0050

Line 5 S2 approach pipe slope, Format F10.4

VVVVO. 5000

Line 6 S2 approach pipe slope, Format F10.4

VVVVO. 7070

Line 7 S2 approach pipe slope, Format F10.4

VVVVO. 8660

Line 8 SHAPE Format 13

VV1

Line 9 B, RM, QMIN , DQ, N1 , N2

Repeat Line 2

Line 10 SHAPE

WO - indicates end of file.

A12.3
oo

o

oS8AT:-»

HYDSUM CALCULATES VALUES OF HC A NO HN TO DETERMINE

WHETHER JUMP FOHATIQN IS POSSIBLE, ALSO VALUES OF

F*M ARE CALCULATED TO INDICATE WHETHER THE JUMP IS

DROWNED ET PIPE ENTRY.

01 HENS ION OT< 10) ,SLO PE C 100) ,HNP ( 1-C0,10 ) , HC P ( LO ) , F 4 l 00 , l 0 ) INTEGER SHAPE

CONHON/CHI/e ,Q ,G,CON ,S 0 , GAN ,PHO , HCR I T, HNOk rl, ARE A ,PE R ,FP N, E NERG

CO MH ON /CM 2/ SHAPE

R£AD44,7Q2i$HAPE

702 FORMAT C 13)

R£AD44,600)B,RM,0MIN,DQ,N1,N2

600 FORMAT 44FIQ.4,213)

a? i n-QHiN

00 601 K *2 ,

0

QT4K)®QT 4 K — 1 ) DQ

601 CONTINUE

1 *0

00 602 J»1,N1

1

38 1 1 RE AD C 4 , 603 ) S 1 603 FORMAT 4 F10.4

)

SLOPE f I )»S

1

602 CONTINUE

00 604 J*i,N2

1 * 1*1

RE AD4 4,603 I S2

SL OPE I I

S “S2

604 CONTINUE

N3-N1*N2

N4-NL

622 CONTINUE

00 605 1*1, N3

00 606 K* l ,

8

0»QT(K ) /1000.0

G*9 • 8

1

Rrt0®1000.0

GAM*G$RHO

SO-SLOPECI)

CQN»<RH**2 ) /SO

C DETERMINATION OF CRITICAL AND NORMAL DEPTHS.

C CALCULATION OF CRITICAL DEPTH.

UP“8

0N*0 ®0

HC “UP/ 2.0

? CONTINUE

CALL CALC4HC,DLS

If CHCRIT J3,4,5

3 ON^HC

GOTO 6

5 UP “HC

6 HCN* CUP*DN ) /2.

0

IF CABS (CHCN-HC )/HC) .LE .0.001 ) GOTO 8 HC * HCN

GOTO 7

9 HC ®HCN

4 If f HCONT.EO.O.O)HCONT»HC

C CALCULATION OF NORMAL DEPTH.

UP B ON® 0.0

HN«UP/2«0

? CONTINUE

A12.4
IFtHNORM) 10,11,12

10 DN-HN

GOTO 13

12 UP-HN

13 HNN»tUP«-DN ) /2. 0

IF tABS ( (HNN-HN 1/HNl .L£ .0.001 I GOT.Q 14

HN-HNN

" GOTO 9

14 HN-HNN

11 CONTINUE

IFtHN.GE.B) HN-B

CALL CALC(HN,OL)

HNP ( I , K) -HN

rtCPCK)»HC

FtI,K)-FPM

606 CONTINUE

605 CONTINUE

IFtSHAPE.EQ.il WRITE(3,607)B,RM

IFCSHAPE.E0.2) WRITE(3,60B)P,RM

607 FORMAT t 1H1 , IX,//, 5* , 'TABULATED NORMAL FLOW DEPTHS',

1' FOR A RECTANGULAR CHANNEL CF WIDTH *»F10.4,' M. ',

2 ' AND MANNING COEFF. ' » F 1 0. 4 » / ) 60S FORMAT ( 1H1 , IX,///// /, ' TABULATED NORMAL FLOW DEPTHS FOR

i'CIRCULAR CROSS SECTION CHANNEL Of DIAMETER ',F10.4,

2* M. ANO MANNING COEFF. ',F10.4,/I

WRITE(3,609J(CTCK),K»1,8)

609 FORMAT t//*5X, 'FLOW RATE - ',eF10.4,' L/S.',5X,

i'PIPE DIA. OR' ,/,107X, 'WIDTH «. ' ) WRITEt 3,619)

619 F0RMATI/50X, 'NORMAL DEPTH M.'l

WRITEt 3,610 ) SLOPE (1 ) ,t HNPt 1,K) , K-l, 3) ,

8

610 FORMATt 5X,'PIPE S LO P E • , 2 X , / , 7X ,

• t S I N) • / , 5X . 1F10.4«2X«9F10.4,IOX,F10.4)

DO 611 I -2 c N 1 WRITEt 3, 6 12) SLOPE t I ), t HNPt I ,K), K-l, 81,

3

612 FORMAT t5X,F10.4,2X, 3F 10.4, 10X, F 1C. 4 I 611 CONTINUE

I »N4

WRITEt 3,613 S 613 FORMATt//, 56X, •****• ,//)

N5-N4«-l

00 614 I -N5 ,N3

WRITEt 3, 61 2) SLOPE! I ),!HNPtI,K) ,K«i,d),8

614 CONTINUE

WRITEt 3,615 IthCPtKl ,<*1,81

615 FORMAT!//, 5X, 'CRITICAL', 4X,eF10. 4, /,5X, 'DEPTH M.',/>

IFtSHAPE.EQ.DWRITEt 3, 616)6, RM

IF t SHAPE. EC. 2 I WRITEt3,617)B,RM

616 FORMAT f 1H1,1X,//////5X» 'TABULATED VALJES QF F«-n AT NORMAL ' I'DEPTHS IN A RECTANGULAR CHANNEL CF WIDTH ',F10.4,

2* M. AND MANNING COEFF. ',F10.4,/I

617 F0RMAT11H1, IX, //////5X, 'TABULATED VALUES OF H AT NORMAL ' I'DEPTHS IN A CIRCULAR CROSS SECTION CHANNEL DIAMETER ',

2F10.4,' M. ANC MANNING COEFF. ',F10.4,/|

WRITEC3,609)!CTtK)»<-L,8)

WRITEt 3,62 0)

620 FORMAT t/43X , 'F M VALUE AT NORMAL DEPT.i, N. • ) WRITE t 3,610 ) SLOPE! 1) * C Ft 1,K ) ,n*l,6) ,8

DO 618 1-2, N3

IFt I.LE.N4)WRITE t3,4L2)SL0PE!I ) , tFt I , < ),K«1,3) ,8

IF t I .GT.N4 ) WRITE t 3, 512 ) SLOPE (1 ),(FtI,<),K-l,d),B

IFt I .£ Q.N4 )WR I TEt 3,613

)
oooooooooo

613 CONTINUE

READ(A,702 ) SHAPE

IF 4 SHAPE .£0.0) GOTO 621

REA04A, 600)8, RM,GHlN»DO,Nl,N2

GOTO 622

621 CONTINUE

END

SUeROUTlNE CALC C H,QL

I

SUBROUTINE CALC IS ‘JSEO THROUGHOUT THE PROGRAM TO

DETERMINE THE FLOW-PIPE PARAMETERS SUCH AS FLOW

DEPTH, AREA, WETTED PERIMETER AS WELL AS BEING USED

IN THE BISECTION METHOD CALCULATION OF NORMAL AND

_ CRITICAL DEPTHS IN EACH OF THE PIPE LENGTHS.

•IN THE CIRCULAR PIPE CROSS SECTICN CASE IT ALSO

CALCULATES SUBTENDED ANGLE AND THE WATER SURFACE

WIDTH AS DEPTH CHANGES.

AS IN BOUND AND MAIN PROGRAM THE PIPE SHAPE IS DETERMINED

BY THE VALUE GF THE TERN SHAPE INPUT AS DATA.

INTEGER SHAPE

COHNON/CHl/e,C,G,CON,SO,GAH,RHO,HCR I T , HNOR M , ARE A , P E R , FP M , E NERG

COMMON/CM2/SHAPE

CONMQN/CM3/ I 2

1 f C SHAPE «GT® 1JGGTO l IF< IZ.EQ.3.AND.H.G6.B)H*8

ARE A »H *0

PER*B»2.Q*H

HCR I T* 1.0-40** 21 *6/ C G* AREA**3)

r|NQRN»1.0-4C**2 )*CON/4 (AREA**3.333)/(PER** 1.33 3) ) DL»HCRIT/(hNORM*SOS

FPH*4GAM*APE A*H/2.0 ) ( RHQ*Q*C/ AREA)

6NERG*H*( Q**2) /( C ARE A* *2 > *2.C*G>

GOTO 2

1 R*B*0.5

PI»3ol^2

IFUZ.EQ.3.ANC .H.GE.B) GOTO 20

IF CH.lT.R I THETA«2.0*ATAN(SCRT4H*48-H) ) / 4B/2.0-H)

)

IFIH.EQ.R ) THETA*PI

IF 4 H.GT.R i THE TA*P I *2.0* A TAN 4 C H-e /2 . 0 I / 4 SGR I C H* ( B-rt i ) )

)

GOTO 22

20 H*0

THE T A* 2 • 0$ P I AR£A*PI*4B/2.0)**2

PER*PI A

0

<0*6/2.

0

GOTO 21

22 CONTINUE

AREA*! ( B**2)/8 .0)*( THE T A-S I N 4 T HE TA )

)

PER*BA THETA/2.0

T»2.0*4 4H*4e-H) )**0.5)

HCR IT* 1.0- 40 **2.0 )*T/( G* ARE A** 3)

HNORM* 1 • 0— 4G**2.0)*CJN/( <AREA**3.333)/4PER**1.333))

DL-HCRIT/(HNORM*SO)

X0*C2. D/3.C)*C 8/ 2 . D ) *4 3 . 0* S I N( THE T A/2 . 0

) -S I N 4 3 . 0*T HE TA/ 2 . 3 J)

1/44.0* (THETA/2 . 0—0. 5*SIN(THETA ) ) ) 21 HBAR-XO+H-e/2.0

FPH*GAM*ARE A*h8AR*-RH0*Q*Q/AREA

ENERG-H*4Q**2) / 4 4 ARE A* *2) *2.0*G

)

2 CONTINUE

RETURN

c N 0

1 BEND

A12.6
NBS-114A (REV. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 81-2367

2. Performing Organ. Report No 3. Publication Date

November 1981

4. TITLE AND SUBTITLE

PREDICTION OF THE HYDRAULIC JUMP LOCATION FOLLOWING A CHANGE OF SLOPE IN A

PARTIALLY FILLED DRAINAGE PIPE

5. AUTHOR(S)

John A. Swaffield

6. PERFORMING ORGANIZATION (If joint or other than N BS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8 . Type of Report & Period Covered

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

Department of Housing and Urban Development

451 7th Street, SW

Washington, D.C. 20410

10. SUPPLEMENTARY NOTES

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

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a si gn ifi cant bibliography or literature survey, mention it here) The criteria governing the formation of a hydraulic jump in a partially filled

fluid conduit downstream of a slope change are presented together with the

necessary techniques to enable water surface profiles and jump location to be

predicted.

Computer programs designed to model the conditions leading to jump formation

under flow and channel scale conditions compatible with current drainage system

design are presented.

The results of a wide range of test conditions in terms of jump formation and

position downstream of a change in channel slope are presented together with a set of criteria to be used in evaluating whether a jump will occur for a given

set of design conditions.

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

building drainage; gradually varied flow; hydraulic jump; partially filled pipe

flow.

13. AVAILABILITY

[Xj Unlimited

| J For Official Distribution. Do Not Release to NTIS

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

20402.

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

14. NO. OF

PRINTED PAGES

193

15. Price

$15.50

USCOMM-DC 6043-P80


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