UNSTEADY FLOW IN OPEN CHANNELS

Transcription

1 UNSTEADY FLOW IN OPEN CHANNELS Definition in free surface flow of water is classified as steady or unsteady flow. The flow of water in rivers, canals, reservoirs, lakes, pools, and free- surface flow in storm water drains, conduits, pipes, galleries, tunnels and culverts, in which the velocities change with time, is defined as unsteady flow ( non - permanent, non - stationary, or time -variable free- surface water flow). Flow in natural channel is always unsteady. When the discharge changes slowly with time is unsteady flow and is approximated by steady flow. The discharge hydrographs in natural streams are largely comprised of using limb followed by recession limb. Those flows occurring during a prolonged drought or those occurring for short time intervals at the highest and lowest points of the hydrographs may be approximately considered to be steady flows. In hydraulic engineering problems it is important to know when to treat an unsteady flow as steady flow. For practical purposes, the answer is obtained by judgment rather than by definite mathematical or experimental criteria. Table: Criteria for classification of Criteria Rate of variation Controlling force Frequency of occurrence Classification Surges (Moving hydraulic jugs - hydraulic bore) Intermediate s Long s capillary gl Gravity C = π Capillary Gravity and friction Simple solitary C = ( ) gl π + πσ ρl tanh πy ρl C = gl (for deep ) π + ( πσ ρl ) 3a η = asech x ct 3 y (for shallow water s) ( ) ( Single form of gravity )

2 a Undisturbed water flow η y Mutiple Wave train The Solitary Wave cg c Direction of movement Relative to channel bed slope Wave Groups Downstream Upstream y V V1 y 1 y 1 V1 V y y 1 V1 V y y 1 V1 V y Type A - Advancing downstream (positive surge) = c + V 1 Type B - Advancing upstream (positive surge) = c - V 1 Type B - Advancing upstream (positive surge) = c + V 1 Type B - Advancing upstream (positive surge) = c - V 1 V y w-v V y w- + V - V V 1 V y y1 y w+ V1 V w-v y y + V 1 V w+ V 1 y1 1 1 Type A Type B Type C Type D Four Types of rapidly varied uniformly progressive flow. (Top) Unsteady flows; (bottom) the corresponding flows that appear steady to an observer following the front. Relative to underlying flow Downstream Upstream RIVER RIVER sea

3 Wave surface elevation Relative to level of underlying flow Surge Positive(higher) uniformly progressive (sluice gate) Negative(lower) Type C (Sluice gate) and D is operated Demand Surge Positive-negative (when Sluice is operated these these appear in pairs) Type A, B occur in Tidal rivers Occur Tidal in rivers Rejected Surge Form Single-faced (Monoclinical progressive rising ) Two-faced Symmetrical Asymmetrical Q FLOOD WAVE Time Periodic or oscillatory Mean Water Level η a L y Translator Orbital Definition Sketch for Oscillatory Wave Motion Particles constantly progressing the movement (Example: Sea s) Transverse Longitudinal Circular particle Orbits in Deep Water

4 Deep water Stokesian Cnoidal Shallow water y y' L Mean water level Elliptical Particle Orbits in Shallow Water Mean water level H Mechanical Oscillation (pulsating flows) Kinematic Diffusive Dynamic Rapidly varying unsteady flow Gradually varying unsteady flow Mechanical oscillation Progressing Surges, bores, depression Typical Cnoidal Wave Profile Floods in reservoirs and in channels / rivers Finite amplitude due to initial disturbance Surface instability Eg: Roll (progressive train) Steep slopes F >.0

5 Ocean Waves and Tides Introduction Wave: An oscillatory movement in a body of water manifested by an alternate rise and fall of the surface. Waves are a most conspicuous feature of the planet ocean. Their sheer size and vigor have always impressed watchers. The scientific study of the s began in the early nineteenth century when Franz Gerstner, proposed to explain the phenomenon on s. According to him, water particles in a move in circular orbits. In 185, Ernst and Wilhelm Weber, in making experimental observations of a tank, concluded that s are reflected without loss of energy. In the twentieth century, oceanographers such as Harold U. Sverdrup and Walter Munk undertook detailed study of s in order to predict and surf movements for naval operations during World War II. Wave parameters: Wave period: The time for a crest to traverse a distance equal to one length. (1) Period (the time it takes two successive crests to pass a fixed point), () length (the distance between two consecutive crests), (3) height (the vertical distance between a trough and a crest). The speed of a moving can be determined as follows: length (L) speed of (C) = period (T) Major components of a typical are depicted in (figure). Wavelength Crest Wave height Trough Wave parameters

6 MECHANISM OF WAVE FORMATION The effectiveness of wind in generating depends on three factors: (1) its average speed, which determines its force, () its duration; and (3) the extent of open water across which it blows (the fetch). When gusty winds blow for a long time and cover large extents of the open water, s of great height (sometimes up to 0 meter) can result. A pressure transducer is a pressure sensing device equipped with a sensitive strain gauge (or potentiometer) that records on a metal diaphragm the slightest change in pressure caused by energy and which subsequently transmits it as an electronic pulse. The intensity of s is reflected by the strength of these electrical pulses. The distinction between the motion of form and the motion of the water mass is important. Waves are carriers of energy imparted to them by wind. Water masses are not. In deep water, forms continue to move forward; but water masses (or the water particles) are except for a slight amount of forward movement, essentially stationary. When a is in deep water, the motion of individual particles at the surface follows a circular orbital pattern and the orbital radius falls off quickly with depth. For example, at a depth equal to one-half the length, the orbital radius is reduced to 4 percent of its surface value. As a result, the water motion gyrates to and fro instead of circularly, and the speed of the water particles decreases rapidly with depth. This mechanism can be illustrated by placing a tennis ball on a water surface. When a wind - produced passes by, the ball will follow a circular orbital movement, bouncing up and down without moving forward. Another ball just below the surface of the water will behave in the same manner but will have a smaller radius to its circular orbit. Sea: Generally chatoic s produced by wind. Swell: Long period s s (as opposed to short period s that are characteristic of a storm). Surf: The breaking s in a coastal region. All unsteady flows involve movements of masses of fluid relative to the distortion and so a problem of kinematics always arises in analyzing them. Indeed, motion or translation of one part of the fluid relative to another is striking feature of all such motions.

7 Waves on the surface of deep water progress at a speed which varies with the length, and are thus called dispersive (Capillary s are similar to light or sound which are non-dispersive). An oscillatory with a sinusoidal water surface profile. If the height 'h' of the is very small compared with length L. The s progress at a speed c given by C = ( gl/π ) tanh ( π y/l) which tends to C ( gl/π) = in deep water, when y /L becomes large: the internal motions of particles are circles whose radius r decreases rapidly with distance z below still water level, according to the exponential law where r = a exp (k/g) where k = π / L and a is the surface radius = h / : the energy of the system is ρg h per unit area of water surface, and in deep water this energy moves at a 'group' speed of c /. A standing system is thus set up in which the vertical motions at places one length apart are double those of either incident or reflected s, yet the motions at places midway between are completely cancelled out. The motions at the places of high amplitude give peculiar vigorous motions and sharp crested s, sometimes called clapotis. The motion that occurs in lakes, excited by wind fluctuations, and is called a seiche. Water orbit motion Direction of motion Arrows indicate instantaneous water motions when is in above position Deep water osicllatory s When these s of infinitesimal height are in relatively shallow water, i.e. when y / L becomes small, the motion under one crest becomes more and more independent of

8 that under the adjacent ones, which for all practical purposes need not be present. The s are then solitary or long s which progress at a velocity c = ( gy ) 1/. A single solitary can exist by a single suitable impulse given to the water at one place. The single forward motion is sometimes termed a translation and the a translatory : this name, as already mentioned, is a poor one since some degree of translation is necessary in any. The s change shape and gradually spread, losing height in the process. No vertical motion at this point any time b length L = b A standing in a water body Two mode b length L = b A seiche in which there is only a half a length at any one instant in the water body In the intermediate range of depths when neither the deep water speed C ( gl/π) =, nor shallow water solitary speed C = gy are applicable. It is found that at low values of the parameter hl / y 3 the two tendencies balance and the s neither spread nor steepen; they are propagated without change of shape, which is that of a complicated elliptic function, and are called cnoidal s. At high values of hl / y 3 the steepening of occurs and the crest gradually steepen until the break.

9 Solitary Still water Level Cnoidal Sinusoidal Three types of s Surges Hydraulic jump - Steady state Type - I Type - II Type - III Type - II s is caused by sudden increase in depth at the downstream end of flow similar to the rising tide into an estuary. This is known as Moving hydraulic jump or Bore Type - I is caused by a sudden reduction of the flow in a channel when gate is closed or discharge is reduced. The front becomes less marked as it progresses away from the Gate and finally dies out in a series of Cnoidal s SURGES - MOVING (TRAVELLING FRONTS) Type - III s is caused by sudden increase of the discharge such as opening of Gates or Dam break Solitary s affected by backward flow and breaks Cnoidal s Sinusoidal (small amplitude s) WAVES IN THE SEA