Lec-5 Unsteady Flow In Open Channels.pdf

Unsteady Flow Surges in Open Channel

Dr. Shahid Ali Professor & HOD Civil Engineering Department NUCES-FAST Lahore

Types of surges

The rapidly-varied transient phenomenon in an open channel, commonly known under the general term surge, occurs wherever there is a sudden change in the discharge or depth or both. Such situations occur, for example, during the sudden closure of a gate. A surge producing an increase in depth is called positive surge and the one which causes a decrease in depth is known as negative surge. Further, a surge can travel either in the upstream or downstream direction, thus giving rise to four basic types. Positive waves generally have steep fronts–sometimes rollers also –and are stable. Consequently they can be considered to be uniformly progressive waves. When the height of a positive surge is small, it can have an undular front. Negative surges, on the hand, are unstable and their form changes with the advance of the surge. Being a rapidly varied flow phenomenon, friction is usually neglected in the simple analysis of surges.

/1.52)^

Problem 1 .

Example A 3.0-m wide rectangular channel has a flow of 3.60 m3/s with a velocity of 0.8 m/s. If a sudden release of additional flow at the upstream end of the channel causes the depth to rise by 50 percent, determine the absolute velocity of the resulting surge and the new flow rate. Solution The flow is shown in Fig. (a). The surge moves in the downstream direction and the absolute velocity of the wave Vw is positive. By superposing (−Vw) on the system the equivalent steady flow is obtained (Fig. (b)).

For a positive surge moving downstream in a rectangular channel.

Example A rectangular channel carries a flow with a velocity of 0.65 m/s and depth of

1.40 m. If the discharge is abruptly increased threefold by a sudden lifting of a gate on the upstream, estimate the velocity and the height of the resulting surge. Solution The absolute velocity of the surge is Vw along the downstream direction. By superimposing a velocity (−Vw) on the system, a steady flow is simulated as shown in Fig.

For a positive surge moving in the downstream direction.

Height of surge

Positive Surge Moving Upstream Figure shows a positive surge moving upstream. This kind of surge occurs on the upstream of a sluice gate when the gate is closed suddenly and in the phenomenon of tidal bores. The unsteady flow is converted into an equivalent steady flow by the superposition of a velocity Vw directed downstream. As before, suffixes 1 and 2 refer to conditions at sections of the channel before and after the passage of the surge, respectively.

(a)Positive surge moving upstream (b) Simulated steady flow

Continuity Equation Momentum equation

From Eqs, two of the five variables y1 , y2 , V1, V2 and Vw can be determined if the three other variables are given. It is to be remembered that in real flow Vw is directed upstream. The velocity V2 however may be directed upstream or downstream depending on the nature of the bore phenomenon.

Moving Hydraulic Jump The Type-1 and Type-2 surges viz. positive surges moving downstream and moving upstream respectively are often termed moving hydraulic jumps in view of their similarity to a steady state hydraulic jump in horizontal channels 1. For Type-1 surge (surge moving downstream) Vr1 = (Vw–V1) 2. For Type-2 surge (surge moving upstream) Vr1 = (Vw+V1)

Energy Loss in Hydraulic Jump

Moving hydraulic jump

RAPIDLY VARIED UNSTEADY FLOW – NEGATIVE SURGES

Celerity and Stability of the Surge The velocity of the surge relative to the initial flow velocity in the canal is known as the celerity of the surge, Cs. Thus for the surge moving downstream Cs=Vw–V1 and for the surge moving upstream Cs=Vw+V1. From above Eqs. it is seen that in both the cases

For a wave of very small height y2 → y1 and dropping suffixes, 𝐶 = √𝑔𝑦 a result which has been used earlier. Consider a surge moving downstream. If the surge is considered to be made up of a large number of elementary surges of very small height piled one over the other, for each of these 𝑉𝑤 = 𝑉1 + √𝑔𝑦 . Consider the top of the surge, (point M in Fig.). This point moves faster than the bottom of the surge, (point N in Fig.). This causes the top to overtake the lower portions and in this process the flow tumbles down on to the wave front to form a roller of stable shape. Thus the profile of a positive surge is stable and its shape is preserved. In a negative surge, by a similar argument, a point M on the top of the surge moves faster than a point on the lower water surface (Fig.). This results in the stretching of the wave profile. The shape of the negative surge at various time intervals will be different and as such the analysis used in connection with positive surges will not be applicable.

For channels of small lengths, the simple analysis of a horizontal frictionless channel gives reasonably good results. However, when the channel length and slope are large, friction and slope effects have to be properly accounted for in a suitable way. Further, changes in the geometry, such as the cross-sectional shape, break in grade and junctions along the channel influence the propagation of surges. A good account of the effect of these factors is available in literature.

Elementary Negative Wave Since the shape of a negative surge varies with time due to the stretching of the profile by varying values of Vw along its height, for purposes of analysis the negative surge is considered to be composed of a series of elementary negative wavelets of celerity √gy superimposed on the existing flow. Consider one such elementary negative wave of height δy .The motion is converted to an equivalent steady-state flow by the superimposition of a velocity (–Vw ) on the system. The resulting steady flow is indicated in Fig. The continuity and momentum equations are applied to a control volume by considering the channel to be rectangular, horizontal and frictionless.

Continuity Equation

(a) Elementary negative wave (b) Equivalent steady flow

Momentum Balance By applying the momentum equation to a control volume enclosing the Sections 1 and 2 in the direction of equivalent steady flow

Introducing the notation as above and neglecting the product of small quantities the momentum equation simplifies to

C = celerity of the elemental at negative wave.

Above equation is the basic differential equation governing a simple negative wave which on integration with proper boundary conditions enables the determination of the characteristics of a negative wave.

Negative Wave Moving Downstream Consider a sluice gate in a wide rectangular channel passing a flow with a velocity of V1 and a normal depth of flow of y1 in the channel downstream of the gate. Consider the sluice gate to partially close instantaneously. Let the new velocity and depth of flow at the gate be V0 and y0 respectively. The closure action of the gate would cause a negative wave to form on the downstream channel (Type 3 wave) and the wave would move in the downstream direction as shown in Fig. The velocity V and depth y at any position x from the gate is obtained by integrating the basic differential equation of a simple negative wave. For the negative wave moving downstream, positive sign is adopted and the resulting basic differential equation is

If the gate movement is instantaneous at t = 0, with reference to the co-ordinates shown in Fig., Vw is in the direction of positive x and hence the profile of the

Equation is the expression for the profile of the negative wave in terms of x, y and t. This equation is valid for the values of y between y0 and y1. Substituting in Eq., the value of gy obtained previous eq. from,

Negative Wave Moving Upstream Figure shows a negative surge produced by instantaneous raising of a sluice gate located at the downstream end of a horizontal, frictionless channel. Type-4 negative wave which starts at the gate is shown moving upstream. Integrating the basic differential equation, The relationship between the velocity and depth is obtained as

Using suffixes 1 and 2 to denote conditions before and after the passage of the wave respectively, and using the boundary condition V = V1 at y = y1

Note that the negative sign of Eq. has been used in deriving Eq. This is done to obtain positive values of V for all relevant values of depth y. The celerity of the wave C in this case is

With reference to the co-ordinate system shown in Fig., the wave velocity Vw is negative in major part of the wave and positive in the lower depths. Considering

The profile of the negative wave is given by

Dam Break Problem A particular case of the above Type-4 negative surge is the situation with V1= 0. This situation models the propagation of a negative wave on the upstream due to instantaneous complete lifting of a control gate at a reservoir. This ideal sudden release of flow from a reservoir simulates the sudden breaking of a dam holding up a reservoir and as such this problem is known as Dam Break problem. Figure shows the flow situation due to sudden release of water from an impounding structure. This is a special case of Type-4 wave with V1 = 0. The coordinate system used is : x = 0 and y = 0 at the bottom of the gate; x is positive in the downstream direction from the gate and negative in the upstream direction from the gate; y is positive vertically upwards.

Dam break Problem

The velocity at any section is

The water surface profile of the negative wave is The profile is a concave upwards parabola. The conditions at the gate are interesting. At the gate, x = 0 and using the suffix 0 to indicate the values at the gate , Note that y0 is independent of time and as such is constant. The salient features of the wave profile are as follows:

The velocity at the gate V0 by The discharge intensity which is also independent of time t. Note that the flow is being analyzed in a horizontal frictionless channel and as such the depth y1 with V1= 0 represents the specific energy, E. At the gate axis (x = 0) Critical depth

Also at x = 0, the Froude number of the flow

Thus the flow at the gate axis is critical and the discharge maximum. Further, it is easy to see that upstream of the gate the flow is sub-critical and on the downstream of the gate (for positive values of x) the flow is supercritical.

This simple ideal analysis of a sudden release from an impounding structure is found to give satisfactory results for a major part of the profile. However, in real situation the downstream end is found to have a rounded positive wave instead of the parabolic profile with its vertex on the x-axis. In actual dam break the tapered leading edge of the ideal profile is modified due to action of ground friction to cause a positive surge to move downstream.

Partial lifting of Downstream Gate

A variation of the dam break problem is the case of partial instantaneous lifting of the downstream gate from initial closed position. A simple case of a sluice gate in a rectangular channel of width B is analyzed as follows. Consider the sluice gate to be suddenly raised by an amount a from an initial closed position. If 4 9

𝑎 ≥ 𝑦1 , then it amounts to full raised position as indicated in the previous section and the 4

analysis is that of the dam break problem. However, if 𝑎 < 𝑦1 , then it is partial closure and 9

an analysis for such a case is given below.

Partial lifting of downstream gate

Refer to Fig. Before the operation of the gate, the water upstream of the gate is at rest at a depth yl. The gate is lifted instantaneously, and partially, so that h0 = drawdown at the gate. A negative wave produced by this action travels upstream with a wave velocity Vw and a forward flow velocity V is created. Since V1 = 0,

At x = 0 y0 = ( y1 –h0 ) and velocity V= V0 The discharge Q0 = By0V0 which is constant as V0 and y0 do not change with time. Q0 can be expressed in terms of h0 and y1 for substituting for y0 and V0, as

On simplification, an expression for the discharge can be obtained in non-dimensional form as

Wave Profile: The profile of the negative wave at any time t is given as

where y = depth of flow at any (x, t).

Example

Example

Example

Example