Analysing Water Surface Profiles

8/10/2019 Analysing Water Surface Profiles

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Water Surface Profil es in GVF (Rectangul ar Channels ONLY)

There are 12 types of water surface flow profiles in GVF. We need a logical classification scheme to identify these profiles.

In general, any problem of varied flow, no matter how complex it may appear, can be brokendown into reaches such that the flow within any reach is either uniform or falls within one ofthe given GVF profiles.

We analyze the stream one reach at a time.

Differentiating (1), the rate of energy dissipation (with = 1) is

21 ( )2

dH dz dy d V dx dx dx g dx

= + + (6)

The last term in (6) can be written as

2 2 2

2 31 ( ) 1 1

2 2d V d q q dy

g dx g dx y g y dx = =

(7)

Also, S = - dH/dL - dH/dx, and S o = -dz/dx, and substituting (7) in (6), we get

2

31ody q

S S dx gy

= +

(8)

or

2 3 2 21 1 1o o oS S S S S S dy

dx q gy V gy Fr

= = =

(9)

Eq. (9) is called the Gradually Varied flow Equation . We can learn a lot about the watersurface profile by studying the terms in (9).

1. dy/dx: If dy/dx is positive, it means the water depth will be increasing along thechannel, and vice versa.

2. Numerator (S o S) [ y compare to y o]: For a wide and shallow rectangular channel, V= q/y and R h = y and from Manningsequation, we get for the nonuniform flow,

For nonuniform flow:

2

5 3

nqS y

=

For uniform flow:

2

5 3oo

nqS

y

=


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2

Comparing these two equations,10 3

o

o

yS S y

=

(10)

3. For constant q and n, (10) shows that when y > y o, S < S o, the numerator (S o S) in(9) will be positive. Conversely, when y < y o, S > S o, (S o S) is negative.

4. Denominator (1 Fr 2) [ y compare to y c]: If Fr = 1 (i.e., y = y c) , the denominator is zero and dy/dx = . If Fr >1 (supercritical, y< y c), the denominator is negative, and if Fr < 1 (subcritical, y > y c), the denominatoris positive.

5. Summary: We note that the signs of the numerator and denominator of (9) can befound fro any depth by comparing it with y o and y c. these 2 signs together give thesign of dy/dx, which in turn defines the slope of the water surface.

we are now ready to study the 12 types of water suface profiles (see Fig. 2), andeventually learn how to sketch them, based on the above analyses.

We will follow a step-by-step procedure to sketch these profiles (not found in text,devised independently and so far, is fail-proof in getting the profile, we shall seethat!).


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Fig. 2 Various types of water surface profiles, flow from left to right

STEP 1: Slope?

Question: What type of bed slope do we have in each reach?Definition: Channel that slopes downward streamwise is positive, i.e., M, S, C and slopesupwards is negative, i.e., A. Horizontal (H) slope is special case with S o = 0.

Action: We need only to compare 2 calculated depths y o and y c , both should becalculated FIRST before we do anything else (unless they are given in the problem). Hence,

(i) M slope is mild if yo > y c (ii) S slope is steep if yo < y c (iii) C slope is critical if yo = y c (iv) H slope is horizontal if S o = 0 (obvious and will always be given)(v) A slope is adverse if S o is negative (only one in this category and is obvious).

Therefore you only need to remember (i) and (ii), and the rest are obvious.

STEP 2: Contro l Points ?

Question: Where are the control points (or sections) in the various reaches? Action: You must locate all the possible controls in the various reaches. This is a bit trickyand it is best to understand the fundamentals of control points first, then try to memorize thelocations of these points!

What is a control point? A control point is a section where there is a definite relationshipbetween the depth and discharge.


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Why do we need control points? Basically, the equations of motion (Navier-Stoke) for ope-channel flow, ecsept for uniform flow, are differential equations, and the solutions requireinitial and boundary conditions to solve. The boundary conditions are specified at channelsections, termed control point (or section).

Where are the locations and types of control points? The theories of disturbancepropagation and of channel transition can answer this question. We will not do this in detail,but merely state some physical facts to allow us to understand the meaning of the termsupstream control and downstream control . These conditions dictate the direction tosketch the profile from the control point (Step 3).

There are 3 types of control points:(1) Uniform-Depth Control (UDC): i.e., Mannings eq. (Q versus y o). Remember, all

profiles would eventually approach uniform or normal flow condition, unless there areother controls that produce a depth other than y o. The location of an UDC is: (i) at theupstream end of the reach in subcritical flow (with mild slope), and (ii) at thedownstream end of the reach in supercrical flow (with steep slope).

(2) Critical-Depth Control (CDC): i.e., critical depth eq. (q versus y c). The location of aCDC is at (i) the choke of a channel transition, eg. hump, constriction, and (ii) the

downstream end of a subcritical-flow regime.(3) Artificial-Channel Control (ACC): The location is in the vicinity of a control structureand is known empirically. For example, , the discharge formula of a sluice gate hasthe form: Q = [2g (y 1 y 2)]0.5 . Thus the relationship is Q versus y 1 y 2, where y 1 andy2 are the depths upstream and downstream of the gate, respectively (since thespecific energy is the same, this means y 1 is the alternate depth of y 2, ref: E-yspecific energy diagram) . Note here there are 2 depths to make up the formula,hence both y 1 and y 2 are control points for a sluice gate.

Why you need to know the meaning of upstream or downstream control? Basically, you wantto know what direction (upstream or downstream) to sketch from. As a rule, the appropriatedirection to begin sketching is towards the upstream for subcritical flow and downstream forsupercritical flow . In other words, you should look for the location of the control point at the

upstream reach for supercitical flow and at the downstream reach for subcritical flow.

STEP 3: Who Control?

Question: Where do we start the sketching from and how? Action: We need to compare two depths here: y and y c (check Step 1 again and note thedifference). What is y? Think of it as the measured water depth or given depth in the reachyou are analyzing. y may be less, equal or greater than y c. If

(i) y > y c Fr < 1, subcritical flow, and we say this is a downstream control case,and you will sketch the profile from the control point (Step 2) towards theupstream direction (you will get a better picture after we go through someexamples).

(ii) y < y c Fr > 1, supercritical flow. This is an upstream control case, and you willsketch the profile from the control point in the downstream direction.

STEP 4: Which Zone?

Question: How many zones are there? And what does zone means? Action: , Basically we are trying to assess where the profile lies (i.e., where y is) with respectto y o and y c. There are 3 zones, consisting of the zone between 3 lines drawn parallel toeach other, i.e., the bed slope line, y o line (YOL) and y c line (YCL). If the stream surface (i.e.,


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depth y) lies above both y o and y c, it is ZONE 1 ; if y is between these lines, it is ZONE 2 ;and if y is below both lines, but above the bed slope line, it is ZONE 3 . The profile must stayinside the zone and cannot cross the 3 lines. Example, If the slope is M and the profile is inzone 1, then the profile is called M1. We have altogether 13 types, as follows:

Slope Zone 1 Zone 2 Zone 3 RemarksM M1 M2 M3S S1 S2 S3C C1 * C3 No C2 cos y o & yc lines mergedH * H2 H3 No H1 cos y o = if S o = 0

A * A2 A3 No A1 cos S o = negative

STEP 5: Type of Profi le?

Question: What type of surface profile would you get based on Steps 1 and 4? Action: Obviously clear, go for it!

STEP 6: Sketch NOW?

Question: Sketch now, but how? Action: You either memorize the shape of the 12 profiles (not easy!) or you learn thefollowing (easier) physical explanation (can also be deduced using Eq. (9)) on how theprofile approaches or cut the y o, y c and the bed slope lines, and the downstream waterdepth.

Join where? Physical explanation on how the water surface profile join the line

Downstreamwater depth

Profile must approach a horizontal asymptote to the downstream water depth (i.e.,control point) - because the velocity is progressively being slowed down withincreasing depth. Examples: M1, S1, C1.

yo line Profile must approach y o line asymptotically - because uniform flow will onlyprevail at sections remote from disturbances (read Sec. 10.1 text). Examples: M1,M2, S2, S3.

yc line Profile will try to cut y c line perpendicularly , theoretically that is because thedenominator of (9) becomes zero in this case. See how the profiles concavetowards y c line, as in M2, M3, S1, S2, C1, C3, H2, H3, A2, A3.

Bed slope line Profile will try to come out of the bed perpendicularly . Examples: M3, S3, H3, C3, A3

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Analysing Water Surface Profiles