Urban Hydraulics. 2.1 Basic Fluid Mechanics

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1 Urban Hydraulics Learning objectives: After completing this section, the student should understand basic concepts of fluid flow and how to analyze conduit flows and free surface flows. They should be able to calculate, - hydrostatic pressure force on a surface - discharge, pressure, energy variations in steady pressurized flows through conduits - force exerted due to steady fluid flow in pipes - impact forces of water jets - energy losses in pipe flows - power requirements for pumping - pressure variation in steady open channel flows - different flow types and energy losses in channel flows - free surface elevation, velocity, depth and discharge in open channel flows - flood propagation in channels They should also be able to derive mathematical formulation for estimation of flood water levels and inundation areas due to floods. 2.1 Basic Fluid Mechanics Definition of a Fluid A fluid is a substance that deforms continuously under the action of an applied shear stress. Thus, a fluid conforms to the shape of its container. The distinction between solids and fluids lies in their behaviour under the action of an external force. A solid when subjected to a shear stress deforms depending on the force and attains a final equilibrium position. Both liquids and gases come under the category of fluids. Liquid retains its own volume but takes the shape of the container. A gas takes the full volume and shape of its container Continuum concept In the analysis of a fluid in civil engineering applications, behaviour and conditions of small fluid volumes containing many molecules are of interest and not of individual fluid molecules. Therefore, fluid is considered as continuous substance. The conditions at a point is the average of a very large number of molecules surrounding the point within a

2 radius large compared to the intermolecular distance. The variation of fluid and flow properties from point to point is considered to be smooth. Any property at a point (x,y,z) at time t can be expressed using space coordinates and time as independent variables, as φ(x,y,z,t). For example, velocity components of v = ui + vj + wk and pressure at a point are expressed as u = u(x,y,z,t) ; v = v(x,y,z,t) ; w = w(x,y,z,t) ; p = p(x,y,z,t) Properties of Fluids a) Density Density ρ is the mass per unit volume where, M is the mass of fluid occupying a volume V. The specific volume of fluid, v s is the volume occupied by a unit mass of fluid and it is also the reciprocal of density. Density has dimensions of ML -3 The specific weight γ = ρg is the weight of the material per given volume. Relative density or specific gravity is the ratio of unit weight of a given fluid to that of water at 4 o C. b) Bulk Modulus Bulk modulus, K is defined as the ratio between volumetric stress and volumetric strain. The negative sign is to account for the reduction in volume due to an increase in pressure intensity, dp. Bulk modulus is an important property of compressible fluids. Bulk modulus, K has the dimension of ML -1 T -2. The bulk modulus of water at 20 C is 2.2 GPa. The bulk modulus varies with pressure and temperature. Density of a fluid will changes with pressure, so all fluids are compressible. However, provided that the changes of density are small, it is often possible to simplify the analysis of a problem by assuming that the fluid is incompressible. Since water is relatively difficult to compress, it is usual to consider water as incompressible or of constant density in the analysis of storm water drainage. However, if the change in pressure is sudden and very large such as in the case of water hammer in pipelines caused by sudden closure of valves, compressibility of the liquid must be taken into account. Under very low pressure conditions that may occur locally in pipelines, water may become vapour. Like any other gas, water vapour is easily compressed and the effects of compressibility and changes of internal energy must be taken into account.

3 c) Viscosity Viscosity is the property of a fluid that enables it to develop resistance to deformation. Newton's law of viscosity: The shear stress in a fluid is proportional to the rate of angular deformation or velocity gradient, or. The proportionality constant µ is known as the dynamic viscosity. A fluid where viscosity is constant with du/dy is called a Newtonian fluid. Dynamic viscosity, µ has the dimension of ML -1 T -1. The viscous resistance offered by liquids is a function of molecular spacing which is in turn a function of temperature. Molecular spacing increases with temperature and hence viscosity of a liquid varies as an inverse function of temperature. Kinematic viscosity ν, is the ratio of dynamic viscosity to density. d) Surface Tension Every molecule exerts upon every other neighboring molecule an electrochemical force known as molecular attraction which gives rise to additional properties of adhesion and cohesion in liquids. At a point not on the surface of a liquid, a molecule is attracted equally in all directions. A molecule at a position very close to the free surface experiences a less force from above it, as the number of molecules above it is less than those below. The resultant force has to act in a direction normal to the free surface. This effect causes the liquid to behave as if it were an elastic membrane under tension. Surface tension is measured as the force acting across unit length of a line drawn in the surface has dimensions of MT -2. In many problems with which engineers are concerned the magnitude of surface tension forces are very small compared with other forces and may be neglected. However, the rise of water in the soil through pores due to the surface tension is of practical significance in soil moisture Fluid statics A fluid is considered static if all the particles of the fluid are motionless. By definition of fluid, tangential or shear stresses are absent when the fluid is at rest. Pressure at a point in a static fluid, with no shear stresses present, is the same in all directions. The system is only under the action of gravitational forces. For an incompressible static fluid, the hydrostatic pressure variation can be written as p = γh where h is the depth measured from the free surface. Pressures at the same level in a

4 continuous expanse of a static fluid are same. e.g. two points at the same elevation in a U-tube manometer. Hydrostatic forces on submerged surfaces Consider a vertical surface that is in contact with liquid on one side (Figure 2.1). The hydrostatic pressure on the surface increases linearly with depth. The sketch of pressure distribution results a pressure prism. For our convenience in analysis, pressure force can be represented as shown by a single force acting at a distance below the free surface. Expressions for and are developed here from a general approach. Figure 2.2 gives the distribution of pressure acting on a submerged surface, inclined plane surface having an irregular cross section. Wall Wall Liquid Z r Pressure destribution R f Figure 2.1 Hydrostatic pressure distribution An element of area has a force acting on it given by The hydrostatic equation, is found in terms of the space variable as Where sin is the vertical depth to. Note that atmospheric pressure acts on both sides and does not affect the resultant force or its location. The element of force acting on becomes

5 p atm θ O p atm Z r O Y O x p a tm R f p atm da z z Figure 2.2 Hydrostatic force on submerged, inclined surface Integration yields the total force exerted on the submerged plane as Therefore, Where is the distance to the centroid of the portion in contact with the liquid as measured from the surface in the direction. To find the location of, let us write an expression for, the moment exerted about point O: The integral on the right-hand side is recognizes as the second moment of area of the submerged plane about the x axis (Figure 2.2) Rather than using, it is more convenient to use a moment of area of the submerged plane. With the parallel axis theorem, we get Then,

6 To determine lateral location in the moment about the axis is, direction, we will follow the same procedure. The By definition, the product of inertia is Using the parallel axis theorem, we get Where, is the second moment about the centroid of the submerged plane and is the distance from axis to the centroid. Then, Fluid flow concepts Conditions in a body of flowing fluid can vary from point to point and, at any given point, can vary from one moment of time to the next. Flow may be classified in many ways based on properties of flow, fluid and geometry involved. A steady flow is one in which the velocity, pressure and cross-section of the stream may vary from point to point but do not change with time. If, at a given point, conditions do change with time, the flow is described as unsteady. In practice, there will always be slight variations of velocity and pressure, but, if the average values are constant, the flow is considered to be steady. If the velocity at a given instant is the same in magnitude and direction at every point in the fluid, flow is described as uniform flow. If, at the given instant, the velocity changes from point to point, the flow is described as non-uniform flow. In practice, when a fluid flows past a solid boundary there will be variations of velocity in the region close to the boundary. However, if the size and shape of the cross-section of the stream of fluid is constant, the flow is considered to be uniform in practice. Therefore, there are, four possible types of flow. (i) Steady uniform flow: Conditions do not change with position or time. The velocity and cross-sectional area of the stream of fluid are the same at each cross-section, e.g. flow of a liquid through a pipe of uniform diameter running completely full at constant velocity. (ii) Steady non-uniform flow: Conditions change from point to point but not with time. The velocity and cross-sectional area of the stream may vary from cross-section to cross-section,

7 but, for each cross-section, they will not vary with time, e.g. flow of a liquid at a constant rate through a tapering pipe running completely full. (iii) Unsteady uniform flow: At a given instant of time the velocity at every point is the same, but this velocity will change with time; e.g. accelerating flow of a liquid through a pipe of uniform bore running full, such condition would occur when a pump is started up. (iv) Unsteady non-uniform flow: The cross-sectional area and velocity vary from point to point and also change with time; e.g. a wave traveling along a channel. Steady or unsteady flow type depends upon the situation of the observer. The frame of reference adopted for describing the motion of a fluid is usually a set of fixed coordinate axes, but the analysis of steady flow is usually simpler than that of unsteady flow and it is sometimes useful to use moving coordinate axes to convert an unsteady flow problem to a steady flow problem. The normal laws of mechanics will still apply, provided that the movement of the coordinate axes takes place with uniform velocity in a straight line Control Volume Concept The conservation laws in their forms familiar to us are directly applicable to systems, i.e, to certain mass of fluid. However, in fluid mechanics, the properties at a given location in the flow and the effect of fluid flow on its boundaries are of more interest than the properties of a fluid mass in motion itself. A formulation of conservation laws applied to a fixed volume in space or control volume approach is thus discussed here. Control volume is defined as a definite volume in space with fixed boundaries through which matter is allowed to cross Reynolds Transport Equation Consider a control volume in an arbitrary flow field as shown in Figure 2.3. At time t, identify a system with its boundary coincident with the boundary of the control volume (Figure 2.3(a)). If N is an extensive property of the flow, we can write where N, N in the system at t, S t N, N in the control volume at t. C t N S, t N C, t At time t + δt, the two boundaries are different as shown in Figure 2.3(b), and we can write

8 N N C, t t S,, t t N in N out System boundary & Control Volume System boundary Control Volume (a) Control volume and system at t (b) Control volume and system at t =t +δt Figure 2.3 : Control volume concept From above two equations, we can write N N N N C, t t C, t S,, t t S, t N in N Dividing by δt and taking the limit δt 0, we can obtain the Reynolds transport equation representing instantaneous rate of change of N, out where dn dt dn dt C S N in Nout - Reynolds transport equation N Rate of inflow of N to the control volume, in N Rate of outflow of N from the control volume, out dn C dt dn S dt Rate of change of N inside the control volume, Rate of change of N in the system Conservation laws applied to fluid flow

9 Mass Continuity Equation Let N be the mass m. Then, by the application of the law of conservation of mass to the system, dm S 0 dt, since the mass in the system is constant. Reynolds transport equation becomes dm dt For steady flow: C m in in m m m out out Ex: Apply continuity equation to find the change in reservoir volume with an inflow and out flow. Force-Momentum Equation Let N be the linear momentum of fluid in X- direction, M X. Then, by the application of the Newton s second law to the system in X-direction, the rate of change of X- momentum is equal to the resultant force in X- direction. The resultant force on the system, dm F XS XS dt. Reynolds transport equation becomes, dm XC FXS M X out M X in dt The resultant force on the control volume constitutes of external forces (or boundary forces) and body forces (e.g.: gravitational force). For steady flow: F M M XS X out X in Ex: Apply force-momentum equation to find the force on a bend of a pipe carrying water. Energy Equation Let N be the total energy (E). Energy can cross a system boundary in the form of heat transfer (to or from the system), in the form of work (done by or on the system), or by mass entering or leaving. When energy crosses the boundary of a system, the energy in the system changes by an equal amount. So a decrease in the energy of a system equals to the increase in energy of the surroundings and vice-versa.

10 Reynolds transport equation becomes de dt C de dt S E in E out The total energy (E) is an extensive property of the system and includes internal, kinetic and potential energies: Total energy per unit mass of fluid, e can be written as Then we can write, where is the average value of total energy per unit mass and is the mass flow rate. If is the change in energy experienced by the system, the law of conservation of energy states that for a system of particles. or where: is the heat added to the system is the work done by the system Then energy equation now becomes, Heat and work are not extensive properties of the system, but they do represent energy transfers across the system boundaries in specific forms. The term consists of all forms of work crossing the boundary including shaft work, electric and magnetic work, viscous shear work, and flow work. The flow work represents work done by the fluid systems in pushing mass into or out of the control volume. Thus where dw ƒ /dt is flow work per unit time. To develop an expression for dw ƒ /dt, consider a mass leaving a control volume through an area da at a unit time as shown in Figure

11 2.4. The fluid in the control volume must work against the external pressure p to move out. With work = force times distance, is the velocity normal to the boundary. Figure 2.4 Sketch for the development of an expression for flow work. We have, Similarly for an element of mass entering, work is being done on the system. Then, Energy equation then becomes For many incompressible flows, is usually assumed to be zero, and changes in internal energy are frequently negligible. A process during which no heat is transferred to or from a system is known as an adiabatic process. Energy equation for steady, incompressible fluid flow becomes,

12 2.1.9 Bernoulli s Equation and its applications The Bernoulli equation gives a relationship between pressure, velocity, and position or elevation in a flow field of an inviscid, incompressible, steady fluid flow and it has many applications in fluid mechanics. Figure 2.5.Control volume in a flow field Consider a control volume containing a length ds of a stream tube (bundle of streamlines) in the flow field (Figure 2.5). Pressure and gravity are the only external forces acting. Evaluating each term separately, we obtain Where is the angle between the direction and gravity. Also, the rate of change in momentum, We get, By the application of force momentum equation, Therefore after simplification, with cos as, ; we get

13 For an incompressible fluid, density is a constant, hence integration along the stream tube gives This is the Bernoulli equation applicable along a streamline in a steady, incompressible, inviscid fluid. It says the total head which is the energy per unit weight of the fluid is constant along a streamline in a steady, incompressible, inviscid fluid flow. Alternatively, the Bernoulli s equation can be developed from the one dimensional energy equation of incompressible flow. When there is no shaft work, viscous work, shear work, electromagnetic work,. In such cases, the energy equation reduces to or Laminar and turbulent flows In laminar flow, fluid particles move along smooth paths in laminas or layers, with one layer sliding smoothly over an adjacent layer. A fluid particle in one layer stays in that layer with no apparent eddies or swirls in the flow. Random fluctuations in particle velocities are damped by the viscous forces and orderly flow is maintained. Laminar flow is not stable in situation involving combinations of low kinematic viscosity high velocity, or large low passages. Fluid particles deviate to move from orderly manner and viscous shear stresses are not sufficient to eliminate the random fluctuations. This type of flow is known as turbulent flow and is characterized by continuous small fluctuations in the magnitude and direction of the velocity of the fluid particles, which are accompanied by corresponding small fluctuations of pressure. Velocity distribution In a real fluid flow whether laminar or turbulent, there is no relative velocity at a contact surface or a boundary, this phenomenon is called no slip condition. Near a stationary wall boundary, there must be a region of flow with a large lateral velocity gradient where the

14 flow velocity increases from zero at the wall. The maximum velocity appears at a distance away from the wall. Typical velocity distributions in pipe flows are shown in Figure 2.6. Increased mixing of flow accompanied by fluctuations in turbulent pipe flow results in steeper velocity gradient at the walls and more even velocity distribution at the central region compared to laminar pipe flow. Even in turbulent flow, there is a thin layer adjacent to the wall where the velocity is small and velocity fluctuations are negligible. This layer in turbulent flows where the thickness depends on the Reynolds number of the flow, exhibits properties of laminar flow and called laminar sub layer. (a ) Laminar flow (b) Turbulent flow Figure 2.6. Velocity profiles in laminar and turbulent flow conditions Shear stress in a fluid flow In laminar flow, shear stresses are developed between layers of fluid moving with different velocities as a result of viscosity. Laminar shear stress is described by Newton s viscosity law which relates shear stress to rate of angular deformation. τ = μ du/dy In turbulent flows, shear stresses are developed between layers of fluid moving with different velocities as a result of viscosity and also as a result of the interchange of momentum due to turbulence causing particles of fluid to move from one layer to another. An equation similar in form to Newton s viscosity law may be written for turbulent shear stress; introducing turbulent eddy viscosity ( η). Turbulent eddy viscosity is a property of the flow and not the fluid. τ = (μ + η) du/dy The velocity of successive layers of fluid will increase as we move away from the boundary. If the stream of fluid is imagined to be of infinite width perpendicular to the boundary, a

15 point will be reached beyond which the velocity will approximate to the free stream velocity, and the drag exerted by the boundary will have no effect. The part of the flow adjoining the boundary in which this change of velocity occurs is known as the boundary layer. In turbulent flow, boundary layer is turbulent boundary layer except in the laminar sub layer adjacent to the wall. The criterion which determines whether the flow is laminar or turbulent is therefore the ratio of the inertial force to the viscous force acting on the particle. If l is a characteristic length in the system under consideration, e.g. the diameter of a pipe, and t is a typical time, then lengths, areas, velocities and accelerations can all be expressed in terms of l and t. For a small element of fluid of mass density ρ, By Newton's second law, Inertial force = Mass x Acceleration α ρl 3. v 2 /l= ρl 2 v 2 From Newton's law of viscosity Viscous shear stress = µ.velocity gradient Viscous force = Viscous shear stress x Area on which stress acts α µ. v/ll l 2. = µ vl It can be shown, that the ratio of inertial force to viscous force = Constant x [( ρvl) / µ] Thus, the criterion which determines whether flow is viscous or turbulent is the quantity [( ρvl)/µ], known as the Reynolds Number (Re). It is a ratio of forces and, therefore, a dimensionless number. For pipe flows, Re is calculated by taking l as the pipe diameter and v as the mean velocity. Experiments carried out with a number of different fluids in straight pipes of different diameters have established that if the Reynolds Number is below a critical value of [( ρvd) / µ]= 2000, flow will normally be laminar. Note: In commercially available pipes used for construction of stormwater conveyance systems flow is generally laminar at values of the Re < 2000, flow is generally turbulent at values of the Re > 4000 and flow is transitional when 2000 < Re < 4000.

16 2.2 Pressurized flow in conduits Pressurized flow (or flow without a free surface) in a conduit is driven by the total head difference at the two ends of the conduit. In a real fluid flow, the total head or the total energy per unit weight of fluid will not remain constant but gradually decrease in direction motion due to losses. Flow through the conduit adjusts to satisfy the condition that the total head loss between two sections is equal to difference in total head at the sections. Consider a flow in a conduit as shown in Figure 2.7. Figure 2.7 Head loss of a flow in a conduit Then,. In a pipe flow, these loses are caused by friction, disturbance to flow at changes in the pipe cross section, and changes in flow direction. Total headline or total energy line referred to the datum is the plot of Hydraulic grade line or piezometric head line referred to the datum is the plot of,

17 In Figure 2.8, the flow of water from the reservoir at A to the reservoir at D is assisted by a pump. At the free surfaces of reservoirs A and D, the fluid has no velocity and pressure is atmospheric pressure or zero gauge pressure. Total head (or total energy per unit weight) in each is represented by the height H A and H A above datum. Water enters the pipe with velocity U 1. There will be a loss of energy due to disturbance of the flow at the pipe entrance and a continuous loss of energy due to friction as the fluid flows along the pipe, so that the total energy line will slope downwards. At B there is a change of section, with an accompanying loss of energy, resulting in a change of velocity to U 2. The total energy line will continue to slope downwards, but at a greater slope since U 2 is greater than U 1 and friction losses are related to velocity. At C, the pump adds energy into the system. The pump develops a head hp, thus providing an addition to the energy per unit weight of hp. So, the total energy line rises by an amount hp. The total energy line falls again due to friction losses and the loss due to disturbance at the entry to the reservoir, where the total energy per unit weight is represented by the height of the reservoir surface above datum. If a piezometer tube were to be inserted at point 1, the water would not rise to the level of the total energy line, but to a level u 2 /2g below it, since some of the total energy is in the form of kinetic energy. Thus, at point 1, the potential energy is represented by Z 1, the pressure energy by p 1 /pg and the kinetic energy by u 2 /2g, the three terms together adding up to the total energy per unit weight at that point. Similarly, at points 2 and 3, the water would rise to levels p 2 /pg and p 3 /pg above the pipe. The line joining all the points to which the water would rise, if an open stand pipe was inserted, is known as the hydraulic grade line or the piezometric head line, and runs parallel to the total energy line at a distance below it equal to the velocity head.

18 Figure 2.8 Energy changes in a pipe flow system Friction losses in pipe flows In most stormwater conveyance systems, the major cause of energy loss is the friction. Other energy losses caused by changes in pipe geometry such as expansion/contraction fittings, bends etc. are called minor losses. Friction loss depends on geometric properties, fluid properties and flow properties in the flow concerned. There are number of semi-empirical or empirical equations established based on experimental investigations to estimate friction loss in pipe flows. Some of them are: Darcy-Weisbach equation, Hazen Williams equation, Mannings equation and Chezy s equation. The first two methods are the commonly used methods and thus only they are described here.

19 The Darcy-Weisbach equation The Darcy-Weisbach equation can be written as 2 L V h L f D 2g Where, h L = head loss due to friction (m) f = f(re, ε/d) is the friction factor Re = ρvd/µ ε/d = relative roughness ε = equivalent sand grain roughness of the pipe (m) L = pipe length (m) D = pipe diameter ( m) V = cross-sectionally averaged velocity of the flow (m/s) g = gravitational acceleration of (m/s 2 ) Friction factor depends on the Reynolds Number (Re = ρvd/µ) of the flow and the relative roughness of the pipe. The relative roughness is defined as the ratio ε/d. Here, the relative roughness value of a pipe increases with aging of the pipe. The slope of the total head line S f (that is, the energy gradient or friction slope) can be expressed as S f hl L 2 1 V f D 2g Experimental investigations by many researchers have established relationships for friction factor in a wide range of flow conditions. The Moody diagram given in Figure 2.9, is a graphical representation of the relationships of f with Re and relative roughness. The Moody diagram can be used to obtain the friction factor f when Re and relative roughness are known. For Reynolds numbers is below 2,000, where the flow is laminar flow, f depends only on the Re. It can be shown that f = 64/Re. For large Reynolds numbers where the flow is fully turbulent or rough turbulent flow, f depends only on the relative roughness of the pipe. In the transitional region between laminar and fully turbulent flow, f depends on both Re and relative roughness.

20 0 Values of (VD) for water at 60 F [Diameter (D) in in., Velocity (V) in ft/sec] Laminar flow, f = 64 Re , h L L V D 2g Relative roughness, e /D Friction factor, f = e, ft. e, mm. Riveted steel Concrete Wood stave Cast iron Galvanized iron Asphalted cast iron Steel or wrought iron Drawn Tubing Smooth pipes (10 3 ) (10 4 ) (10 5 ) (10 6 ) (10 7 ) Reynolds number, Re = VD n Fig. 2.9 Moody diagram The Hazen Williams Equation The Hazen-Williams equation was developed primarily for use in water distribution design. The equation is Where, V = flow velocity (m/s) C f = a unit conversion factor (0.849 for SI units) C h = Hazen Williams resistance coefficient R = hydraulic radius (m) S f = Energy gradient Some typical values of the Hazen-Williams resistance coefficient (C h ) are given in Table 2.1

21 Table 2.1 Typical values of the Hazen-Williams resistance coefficient Pipe Material C h Cast iron (new) 130 Cast iron (20 yr old) 100 Concrete (average) 130 New welded steel 120 Asbestos cement 140 Plastic Minor losses In addition to the spatially continuous energy losses due to friction, energy losses occur at fittings in pipelines such as entrance and exits, reservoirs/man holes, pipe expansion and contractions, changes in pipe alignment. In a long pipeline these minor losses are smaller compared to friction losses. However, the energy loss at a control valve has a primary effect in regulating the discharge in a pipeline. Head loss at a fitting is expressed as 2 V h L K 2g Where, V = velocity at the downstream (m/s) K = loss coefficient Typical values of loss coefficients are shown in Table 2.2. Table 2.2 Typical values of loss coefficients Fitting K Flanged 90 o elbow Globe valve fully open 10 Flange T-joint Line flow Branch flow Sudden expansion, referred to upstream velocity head. D 1 and D2 : upstream and downstream velocities respectively

22 2.3 Open channel flow Open channel flow is a flow with free surface and the pressure on the free surface is normally the atmospheric pressure. Open channels may be manmade or natural. In an urban environment, drains, sewers and streams where flows occur with a free surface, are open channels. They may be prismatic (cross sectional shape does not change along the canal) or non-prismatic, and lined or unlined. A roadside drain with constant slope and cross-section is an example of a prismatic channel. Natural streams are examples of nonprismatic channels. In a lined channel, the perimeter is paved with a material as concrete to protect against erosion by the flow. Unlike in pressurized pipe flow, the cross-sectional area of flow at a given section is not a constant but a function of flow depth. Also, the depth of flow at a section for any specific discharge depends on channel geometry as well as the flow conditions dictated by the boundary conditions at control sections Pressure Variations in Open-Channel Flow Consider flow in a channel of unit width and a constant bed slope of angle θ, shown in Figure dl d p.dl θ Figure 2.10 Pressure Variations Consider a control volume bounded by a flow length dl, the depth d measured normal to the channel bottom. If the pressure at the depth of d is p, the application of forcemomentum equation in the direction perpendicular to the flow, with the assumption that the acceleration of flow in that direction is negligible, we get

23 Slope is small and then, That is, in an open channel flow with small bottom slope and no flow acceleration in the direction perpendicular to the flow, the pressure distribution is hydrostatic Types of open channel flows The motivating force establishing flow is predominantly the gravity force component acting parallel to the bed slope but net pressure forces and inertia forces may also be present. Open-channel flows can be classified into many categories according to how the depth of the flow changes with respect to the time and the position as discussed under Section Flow in channel may be steady or unsteady. Steady flow may be uniform or non-uniform depending on whether or not the mean flow is constant along the channel. Steady uniform flow occurs when the motivating forces and resistance forces are exactly balanced over the reach. The non-uniform open channel flows are further classified into gradually varied flow and rapidly varied flow. Gradually varied flows occur when the flow depth at the control is different from the depth of uniform flow. The depth of flow in a gradually varied flow changes along the length of a channel as a consequence of convective accelerations of the flow. Gradually varied flows are common at the upstream of reservoirs, at channel entrances and exits, and at locations where channel geometric properties change. Flow undergoes abrupt changes within a short channel length in rapidly varied flows, and the assumption of negligible vertical acceleration of the flow hence, the hydrostatic pressure distribution is not warranted. Example of rapidly varied flow is the hydraulic jump in open channel flow Open channel geometry factors The longitudinal slope and the geometric parameters of the channel cross section are important geometric properties of an open channel. The common cross-sectional shapes of prismatic channels are rectangular, triangular, trapezoidal, and circular. The cross-sectional area, wetted perimeter, hydraulic radius, top width and hydraulic depth can be determined based on the depth of flow in a channel section of specific shape. The hydraulic radius is defined as

24 R = A/P Where, R = hydraulic radius ( m) A = cross-sectional area (m 2 ) P = wetted perimeter (m) The definition of hydraulic depth which is the average depth of flow in a cross section, is D h = A/T, Where, D h = hydraulic depth (m) T = top width (m) Specific Energy and Critical Flow Specific energy is the energy head relative to channel bottom elevation. For a channel of small slope, the specific energy E is 2 2 V Q E y y 2 2g 2gA Where, E = specific energy (m) Y = depth of flow (m) α = velocity distribution coefficient V = cross-sectionally averaged flow velocity (m/s) g = gravitational acceleration (9.81m 2 /s) Q = discharge (m 3 /s) A = cross-sectional area (m 2 ) A graph of E vs y for a given discharge Q and channel geometry A = f(y) is called the specific energy curve. For different values of Q, we get a family of E-y curves. As Q increases the curves shift to the right. Specific energy curve is a parabola and shows the variation of E with y for a fixed discharge and channel cross-sectional geometry. For a given specific energy curve, E becomes minimum at a certain depth called the critical depth y c. The flow at this depth is called the critical flow. For this condition, de/dy is equal to zero;