Water Flow in Open Channels

Transcription

1 The Islamic Universit of Gaza Facult of Engineering Civil Engineering Department Hdraulics - ECIV 33 Chapter 6 Water Flow in Open Channels

2 Introduction An open channel is a duct in which the liquid flows with a free surface. Open channel hdraulics is of great importance in civil engineers, it deals with flows having a free surface, for example: Channels constructed for water suppl, irrigation, drainage, and Sewers, culverts, and Tunnels flowing partiall full; and Natural streams and rivers.

3 Pipe Flow and Open Channel Flow Pipe Flow The liquid completel fills the pipe and flow under pressure. The flow in a pipe takes place due to difference of pressure (pressure gradient), The flow in a closed conduit is not necessaril a pipe flow. Open Channel Flow Flow takes place due to the slope of the channel bed (due to gravit). The flow must be classified as open channel flow if the liquid has a free surface. 3

4 Pipe Flow Open Channel Flow 4

5 For Pipe flow (Fig. a): The hdraulic gradient line (HGL) is the sum of the elevation and the pressure head (connecting the water surfaces in piezometers). The energ gradient line (EGL) is the sum of the HGL and velocit head. The amount of energ loss when the liquid flows from section to section is indicated b h L. For open channel flow (Fig. b): The hdraulic gradient line (HGL) corresponds to the water surface line (WSL); where it subjected to onl atmospheric pressure which is commonl referred to as the zero pressure reference. The energ gradient line (EGL) is the sum of the HGL and velocit head. The amount of energ loss when the liquid flows from section to section is indicated b h L. For uniform flow in an open channel, this drop in the EGL is equal to the drop in the channel bed. 5

6 6. Classifications of Open Channel Flow Classification based on the time criterion:. Stead Flow (time independent) (discharge and water depth do not change with time). Unstead Flow (time dependent) (discharge and water depth at an section change with time) Classification based on the space criterion:. Uniform flow (are mostl stead) (discharge and water depth remains the same at ever section in the channel). Non-uniform Flow (discharge and water depth change at an section in the channel) 6

7 Non-uniform flow is also called varied flow ( the flow in which the water depth and or discharge change along the length of the channel), it can be further classified as: Graduall varied flow (GVF) where the depth of the flow changes graduall along the length of the channel. Rapidl varied flow (RVF) where the depth of flow changes suddenl over a small length of the channel. 7

8 a) Uniform flow are mostl stead b) Unstead uniform flows are ver rare in nature c) Stead varied flow (over a spillwa crest) d) Unstead varied flow (flood wave) e) Unstead varied flow (tidal surge) 8

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10 6. Uniform Flow in Open Channel Uniform flow in an open channel must satisf the following main features:. The water depth, flow area A, discharge Q, and the velocit distribution V at all sections throughout the entire channel length must remain constant.. The slope of the energ gradient line (S e ), the water surface slope (S ws ), and the channel bed slope (S 0 ) are equal. S e S ws S 0 0

11 This is possible when the gravit force (W sin θ) component equal the resistance to the flow (F f ) W sinθ + F F Hdrostati c forces at ends sinθ W sinθ F τ 0 f K cns tan t of proportion alit γals F tanθ F S ( γal)sinθ γals τ PL ( KV 0 0 F f ) PL 0 resisting force per unit area of channel, γ A ( KV ) PL V. S K P V C R S h e C γ Chez cons tant R h HdraulicRadius K A P The Chez Formula

12 R h hdraulic radius or hdraulic mean depth area of flow ( wetted area ) R h wetted perimeter C Chez coefficient (Chez s resistance factor), m / /s, varies in relation of both the conditions of channel and flow. Manning derived the following empirical relation: A P C n R / 6 h where n Manning s coefficient for the channel roughness See the next table for tpical values of n.

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14 Manning s formula Substituting into Chez equation, we obtain the Manning s formula for uniform flow: V n / 3 /3 R h S e OR Q VA A R h Se n Where: Q in m 3 /sec, V in m/sec, R h in m, S e in (m/m), n is dimensionless 4

15 Example Open channel of width 3m as shown, bed slope :5000, d.5m find the flow rate using Manning equation, n0.05. V Rh n A 0.5 P 3 S e ( 3 + 9) ( ) m m.5m A 9 Rh P V Q VA m 3 / m/s s 5

16 Example The cross section of an open channel is a trapezoid with a bottom width of 4 m and side slopes :, calculate the discharge if the depth of water is.5 m and bed slope /600. Take Chez constant C 50. 6

17 Example Open channel as shown, bed slope 69:584, find the flow rate using Chez equation, take C35. 7

18 8 ( ) ( ) s VA Q V P A R P A R S C V h e h / 3.84 m m/s m m

19 6.3 Hdraulic Efficienc of open channel sections Based on their existence, an open channel can be natural or artificial: Natural channels: such as streams, rivers, valles, etc. These are generall irregular in shape, alignment and roughness of the surface. Artificial channels: built for some specific purpose, such as irrigation, water suppl, wastewater, water power development, and rain collection channels. These are regular in shape and alignment with uniform roughness of the boundar surface. 9

20 Based on their shape, an open channel can be prismatic or nonprismatic: Prismatic channels: the cross section is uniform and the bed slop is constant. Non-prismatic channels: when either the cross section or the slope (or both) change, the channel is referred to as non-prismatic. It is obvious that onl artificial channel can be prismatic. The most common shapes of prismatic channels are rectangular, parabolic, triangular, trapezoidal and circular; see the next figure. 0

21 Most common shapes of prismatic channels

22 Most economical section is called the best hdraulic section or most efficient section as the discharge, passing through a given cross-sectional area A, slope of the bed S 0 and a resistance coefficient, is maximum. Hence the discharge Q will be maximum when the wetted perimeter P is minimum. Q AV AC R h S e AC A S P e const.* P

23 Economical Rectangular Channel A B D, P D + B A P D + D P should be minimum for a given area; dp dd A A 0 D D A B D D D R h 4 P B+ D D+ D dp 0 dd BD D D D R h B D D D B So, the rectangular channel will be most economical when either: the depth of the flow is half the width, or the hdraulic radius is half the depth of flow. 3

24 A (B + nd )D Economical Trapezoidal Channel P B+ D + n B A P ( nd ) + D + n D dp dd 0 A D nd dp A n+ + n 0 dd D (B+ nd)d B nd + n + n + D D or P B + B + n D ( B + n D ) D A + n D B+ nd + n + n A Rh P ( B + n D ) D ( B + n D ) D Rh 4

25 Other criteria for economic Trapezoidal section When a semi-circle is drawn with the trapezoidal center, O, on the water surface and radius equal to the depth of flow, D, the three sides of the channel are tangential to the semi-circle. To prove this condition, using the figure shown, we have: B OF OM sin α ( B + n D ) sin α ( + n D ) sin α OF sinα B + n D using triangle KMN, we have: MK sinα MN D D + n 5

26 OF ( B ) + n D + n B + n D using equation D + n to replace the numerator, we obtain: OF D + + n n OF D Thus, if a semi-circle is drawn with O as center and radius equal to the depth of flow D, the three sides of a most economical trapezoidal section will be tangential to the semicircle. 6

27 The best side slope for Trapezoidal section when n 3 θ 60 ο D P + n B+ n D B D ( + n n) B + B + n D ( B + n D ) A D n D D + n ( n ) D A + n n B A D n D 7

28 Now, from equations: P ( B + n D) B A P D n D A D squaring both sides dp dn n + n 0 A P 4( ) 4 A( + n n) D P dp 4 A [( + n ) *( n) ] dn 4n + n n 3 3 n tanθ θ ο 60 The best side slope is at 60 o to the horizontal, i.e.; of all trapezoidal sections a half hexagon is most economical. However, because of constructional difficulties, it ma not be practical to adopt the most economical side slopes 8

29 αd A 4 d 8 P α r αd sinα Circular section In the case of circular channels, the area of the flow cannot be maintained constant. Indeed, the cross-sectional area A and the wetted perimeter P both do not depend on D but the depend on the angle α. Referring to the figure shown, we can determine the wetted perimeter P and the area of flow A as follows: Thus in case of circular channels, for most economical section, two separate conditions are obtained:. Condition for maximum discharge, and. Condition for maximum velocit. 9

30 . Condition for Maximum Discharge for Circular Section: Q AV AC R S C A 3 h P S Q C S A P 3 dq dα 0 (Using the Chez formula) α ο 54 D d (Using Manning s formula) α ο 5 D d. Condition for Maximum Velocit for Circular Section: V C R S C A h P S V C S α ο. D d A P dv dα 0 30

31 Variation of flow and velocit with depth in circular pipes 3

32 6.4 Energ Principle in Open Channel Flow The total energ of a flowing liquid per unit weight is given b: Total Energ Z + + V g If the channel bed is taken as the datum, then the total energ per unit weight will be: V E specific + g Specific energ (Es) of a flowing liquid in a channel is defined as energ per unit weight of the liquid measured from the channel bed as datum. It is a ver useful concept in the stud of open channel flow. 3

33 E V + E + g E s p k Es + Q g A E p potential energ of flow E k kinetic energ of flow Valid for an cross section V g Specific Energ Curve: It is defined as the curve which shows the variation of specific energ (Es ) with depth of flow. 33

34 Specific Energ Curve (Rectangular channel) Consider a rectangular channel in which a constant discharge Q q discharge per unit width constant ( since Q and B are constants) V Q A Q B q B q E + E + s p g E p E k E K E P E s c 34

35 Sub-critical, critical, and supercritical flow The criterion used in this classification is what is known b Froude number, Fr, which is the measure of the relative effects of inertia forces to gravit force: F r V gd h Q T A g F r 3 T V mean velocit of flow of water, D h hdraulic depth of the channel D h Area of Flow (Wetted Area) Water Surface Width A T T Fr Fr < Fr Fr > Flow Sub-critical Critical Supercritical 35

36 Referring to the energ curve, the following features can be observed:. The depth of flow at point C is referred to as critical depth, c. (It is defined as that depth of flow of liquid at which the specific energ is E min c minimum, The flow that corresponds to this point is called critical flow (Fr.0).. For values of E s greater than E min, there are two corresponding depths. One depth is greater than the critical depth and the other is smaller then the critical depth, for example; E and ' s These two depths for a given specific energ are called the alternate depths. 3. If the flow depth > c the flow is said to be sub-critical (Fr <.0). In this case E s increases as increases. 4. If the flow depth < c the flow is said to be super-critical (Fr >.0). In this case E s decreases as increases. 36

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38 de d Critical depth, c for rectangular channel Critical depth, c, is defined as that depth of flow of liquid at which the specific energ is minimum, E min, The mathematical expression for critical depth is obtained b differentiating energ equation with respect to and equating the result to zero; E s + q g d 0 d q g ( q q g ) g ( ) q g q c g

39 39 Critical velocit, V c for rectangular channel, 3 g q c q B Q A Q V V q c c OR g V c c c 3 V g c c r c c F g V

40 Minimum Specific Energ in terms of critical depth Emin c + 3 c q g q g c Emin c + c c Emin 3 OR E c 3 min 40

41 Critical depth, c, for Non- Rectangular Channels de d Q Q da s 0 ( + ) ( ) d 0 3 d g A g A d Q da OR ( ) 0 3 g A d (constant discharge is assumed) da/d the rate of increase of area with respect to T (top width). Q T 0 g A 3 Recalling that D h Q g A T 3 A condition must be satisfied for the flow at the critical depth. Q g T A D h The equation ma also be written in terms of velocit V g D h The velocit head is equal to one-half the hdraulic depth for critical flow. 4

42 Q A E s + E s + This equation represents g A T the critical state OR E c A c + ( T ) The general equation for the specific energ in critical state applicable to channels of all shapes. Rectangular section Trapezoidal section E c 3 c E c ( 3B + 5n ) c ( B + n ) c c Circular section Triangle section d d Ec ( + cos α ) 6 ( α sin α ) sinα E c 5 4 c 4

43 Constant Specific Energ The specific energ was varied and the discharge was assumed to be constant. Let us now consider the case in which the specific energ is kept constant and the discharge Q is varied. Q E s Q A g E s g A ( ) s s Q A ( g) ( E ) ga E ga The discharge will maximum if dq d 0 Q dq d g E A da g A da s( ) ( + A ) d d da/d T ge ( AT ) g ( AT ) ga 0 s 43

44 4EsT 4T A 0 T ( E s ) A E A s + T but Es + Q + g A Q g A A + T Q g 3 A T Thus for a given specific energ, the discharge in a given channel is a maximum when the flow is in the critical state. The depth corresponding to the maximum discharge is the critical depth. 44

45 6.5 Hdraulic Jump A hdraulic jump occurs when flow changes from a supercritical flow (unstable) to a sub-critical flow (stable). There is a sudden rise in water level at the point where the hdraulic jump occurs. Rollers (eddies) of turbulent water form at this point. These rollers cause dissipation of energ. A hdraulic jump occurs in practice at the toe of a dam or below a sluice gate where the velocit is ver high. 45

46 General Expression for Hdraulic Jump: In the analsis of hdraulic jumps, the following assumptions are made: () The length of hdraulic jump is small. Consequentl, the loss of head due to friction is negligible. () The flow is uniform and pressure distribution is due to hdrostatic before and after the jump. (3) The slope of the bed of the channel is ver small, so that the component of the weight of the fluid in the direction of the flow is neglected. 46

47 Location of hdraulic jump Generall, a hdraulic jump occurs when the flow changes from supercritical to subcritical flow. The most tpical cases for the location of hdraulic jump are:. Jump below a sluice gate.. Jump at the toe of a spillwa. 3. Jump at a glacis. (glacis is the name given to sloping floors provided in hdraulic structures.) 47

48 The net force in the direction of flow the rate of change of moment in that direction Q γ g ( V V ) The net force in the direction of the flow, neglecting frictional resistance and the component of weight of water in the direction of flow, R F - F. Therefore, the impulse-moment ields F γ Q F ( V V ) g Where F and F are the pressure forces at section and, respectivel. γ A γ A Q ga γ Q γ A ( V V ) g γ Q γ A ( ) g A A Q + A + ga A the distance from the water surface to the centroid of the flow area 48

49 Q ga Q + A + ga A Comments: This is the general equation governing the hdraulic jump for an shape of channel. The sum of two terms is called specific force (M). So, the equation can be written as: M M This equation shows that the specific force before the hdraulic jump is equal to that after the jump. 49

50 Q A ga B using Hdraulic Jump in Rectangular Channels Q + A + ga q Q B A B q g A Q g B Q + ( B )( ) + ( B )( ) g B, we get q + g ( ) q + 0 g 50

51 This is a quadratic equation, the solution of which ma be written as: q + + g + + 8q g 3 q + + g + + 8q g 3 where is the initial depth and is called the conjugate depth. Both are called conjugate depths. These equations can be used to get the various characteristics of hdraulic jump. 5

52 5 q g c c c ( ) 8 F + + F g V F F V g But for rectangular channels, we have Therefore, These equations can also be written in terms of Froude s number as:

53 Head Loss in a hdraulic jump (H L ): Due to the turbulent flow in hdraulic jump, a dissipation (loss) of energ occurs: H L E E E Where, E specific energ For rectangular channels: hence, Es + q g q q HL + g + g 3 After simplifing, we obtain E H L ( 4 ) 3 53

54 Height of hdraulic jump (h j ): The difference of depths before and after the jump is known as the height of the jump, h j Length of hdraulic jump (L j ): The distance between the front face of the jump to a point on the downstream where the rollers (eddies) terminate and the flow becomes uniform is known as the length of the hdraulic jump. The length of the jump varies from 5 to 7 times its height. An average value is usuall taken: L j 6h j 54

55 6.6 Graduall Varied Flow Non-uniform flow is a flow for which the depth of flow is varied. This varied flow can be either Graduall varied flow (GVF) or Rapidl varied flow (RVF). Such situations occur when: - control structures are used in the channel or, - when an obstruction is found in the channel, - when a sharp change in the channel slope takes place. 55

56 Classification of Channel-Bed Slopes The slope of the channel bed is ver important in determining the characteristics of the flow. Let S 0 : the slope of the channel bed, S c : the critical slope or the slope of the channel that sustains a given discharge (Q) as uniform flow at the critical depth ( c ). n is is the normal depth when the discharge Q flows as uniform flow on slope S 0. 56

57 The slope of the channel bed can be classified as: ) Critical Slope C : the bottom slope of the channel is equal to the critical slope. S0 Sc or n c ) Mild Slope M : the bottom slope of the channel is less than the critical slope. S0 < Sc or n > c 3) Steep Slope S : the bottom slope of the channel is greater than the critical slope. S0 > Sc or n < c 4) Horizontal Slope H : the bottom slope of the channel is equal to zero. S ) Adverse Slope A : the bottom slope of the channel rises in the direction of the flow (slope is opposite to direction of flow). S 0 negative 57

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59 Classification of Flow Profiles (water surface profiles) The surface curves of water are called flow profiles (or water surface profiles). The shape of water surface profiles is mainl determined b the slope of the channel bed S o. For a given discharge, the normal depth n and the critical depth c ma be calculated. Then the following steps are followed to classif the flow profiles: - A line parallel to the channel bottom with a height of n is drawn and is designated as the normal depth line (N.D.L.) - A line parallel to the channel bottom with a height of c is drawn and is designated as the critical depth line (C.D.L.) 3- The vertical space in a longitudinal section is divided into 3 zones using the two lines drawn in steps & (see the next figure) 59

60 4- Depending upon the zone and the slope of the bed, the water profiles are classified into 3 tpes as follows: (a) Mild slope curves M, M, M 3. (b) Steep slope curves S, S, S 3. (c) Critical slope curves C, C, C 3. (d) Horizontal slope curves H, H 3. (e) Averse slope curves A, A 3. In all these curves, the letter indicates the slope tpe and the subscript indicates the zone. For example S curve occurs in the zone of the steep slope. 60

61 Flow Profiles in Mild slope Flow Profiles in Steep slope 6

62 Flow Profiles in Critical slope Flow Profiles in Horizontal slope Flow Profiles in Adverse slope 6

63 Dnamic Equation of Graduall Varied Flow Objective: get the relationship between the water surface slope and other characteristics of flow. The following assumptions are made in the derivation of the equation. The flow is stead.. The streamlines are practicall parallel (true when the variation in depth along the direction of flow is ver gradual). Thus the hdrostatic distribution of pressure is assumed over the section. 3. The loss of head at an section, due to friction, is equal to that in the corresponding uniform flow with the same depth and flow characteristics. (Manning s formula ma be used to calculate the slope of the energ line) 4. The slope of the channel is small. 5. The channel is prismatic.. 6. The velocit distribution across the section is fixed. 7. The roughness coefficient is constant in the reach. 63

64 Consider the profile of a graduall varied flow in a small length dx of an open channel the channel as shown in the figure below. The total head (H) at an section is given b: H Z + + V g Taking x-axis along the bed of the channel and differentiating the equation with respect to x: dh dx dz d + + dx dx d dx V g 64

65 dh/dx the slope of the energ line (S f ). dz/dx the bed slope (S 0 ). Therefore, d Sf S0 + + dx d dx V g Multipling the velocit term b d/d and transposing, we get d dx d dx + d d d dx S + 0 V g S f d V d g S 0 S f or d dx + d d V g S 0 S f This Equation is known as the dnamic equation of graduall varied flow. It gives the variation of depth () with respect to the distance along the bottom of the channel (x). 65

66 The dnamic equation can be expressed in terms of the discharge Q: d dx S 0 S f Q T g A 3 The dnamic equation also can be expressed in terms of the specific energ E : d dx de / dx Q T g A 3 66

67 Depending upon the tpe of flow, d/dx ma take the values: (a) d dx 0 The slope of the water surface is equal to the bottom slope. (the water surface is parallel to the channel bed) or the flow is uniform. (b) d dx positive The slope of the water surface is less than the bottom slope (S 0 ). (The water surface rises in the direction of flow) or the profile obtained is called the backwater curve. (c) d dx negative The slope of the water surface is greater than the bottom slope. (The water surface falls in direction of flow) or the profile obtained is called the draw-down curve. 67

68 Notice that the slope of water surface with respect to horizontal (S w ) is different from the slope of water surface with respect to the bottom of the channel (d/dx). A relationship between the two slopes can be obtained: Consider a small length dx of the open channel. The line ab shows the free surface, The line ad is drawn parallel to the bottom at a slope of S 0 with the horizontal. The line ac is horizontal. The water surface slope (Sw) is given b bc Sw sinφ ab cd bd ab Let θ be the angle which the bottom makes with the horizontal. Thus cd cd S0 sinθ ad ab 68

69 The slope of the water surface with respect to the channel bottom is given b d dx bd ad bd ab S w S 0 d dx This equation can be used to calculate the water surface slope with respect to horizontal. d dx S 0 S w 69

70 Water Profile Computations (Graduall Varied Flow) Engineers often require to know the distance up to which a surface profile of a graduall varied flow will extend. To accomplish this we have to integrate the dnamic equation of graduall varied flow, so to obtain the values of at different locations of x along the channel bed. The figure below gives a sketch of calculating the M curve over a given weir. 70

71 Direct Step Method One of the most important method used to compute the water profiles is the direct step method. In this method, the channel is divided into short intervals and the computation of surface profiles is carried out step b step from one section to another. For prismatic channels: Consider a short length of channel, dx, as shown in the figure. dx 7

72 Appling Bernoulli s equation between section and, we write: S dx V V g g 0 S dx f or S dx + E E + S dx 0 f or dx E S E 0 S f where E and E are the specific energies at section and, respectivel. This equation will be used to compute the water profile curves. 7

73 The following steps summarize the direct step method:. Calculate the specific energ at section where depth is known. For example at section -, find E, where the depth is known ( ). This section is usuall a control section.. Assume an appropriate value of the depth at the other end of the small reach. Note that: > if the profile is a rising curve and, < if the profile is a falling curve. 3. Calculate the specific energ (E ) at section - for the assumed depth ( ). 4. Calculate the slope of the energ line (S f ) at sections - and - using Manning s formula V n R / 3 S f and And the average slope in reach is calculated V n R / 3 S fm S S f + S f f 73

74 5. Compute the length of the curve between section - and - L E, dx S E 0 S fm or L, S 0 E S f E + S f 6. Now, we know the depth at section -, assume the depth at the next section, sa 3-3. Then repeat the procedure to find the length L,3. 7. Repeating the procedure, the total length of the curve ma be obtained. Thus L L, + L, L n, n where (n-) is the number of intervals into which the channel is divided. 74