Uniform Flow PDF

Uniform Flow In free-surface flow, the component of the weight of water in the downstream direction causes acceleration of flow (it causes deceleration if the bottom slope is negative), whereas the shear stress at the channel bottom and sides offers resistance to flow. Depending upon the...

Uniform Flow In free-surface flow, the component of the weight of water in the downstream direction causes acceleration of flow (it causes deceleration if the bottom slope is negative), whereas the shear stress at the channel bottom and sides offers resistance to flow. Depending upon the relative magnitude of these accelerating and decelerating forces, the flow may accelerate or decelerate.  If the resistive force is more than the component of the weight, then the flow velocity decreases and, to satisfy the continuity equation, the flow depth increases. The converse is true if the component of the weight is more than the resistive force. If the channel is long and prismatic (i.e., channel cross section and bottom slope do not change with distance), then the flow accelerates or decelerates for a distance until the accelerating and resistive forces are equal. From that point on, the flow velocity and flow depth remain constant. Such a flow, in which the flow depth does not change with distance, is called uniform flow, and the corresponding flow depth is called the normal depth.  Flow Resistance Assume that the flow is one-dimensional i.e. there are no secondary currents in the flow and the shear resistance to flow at the boundaries is uniform. Flow Resistance Equations We first derive an equation for non-uniform flow and then simplify it for uniform flow as a special case of non-uniform flow. Chezy Equation Assumptions: the flow is steady; the slope of the channel bottom is small; and the channel is prismatic. Consider a control volume of length Δx At the upstream side of this control volume, let the distance be x, flow velocity be V , and the flow depth be y. Values of these variables at the downstream side are x+Δx, V+(dV/dx)Δx, and y+(dy/dx)Δx). The following forces are acting on the control volume Pressure force on the upstream side, F1 Pressure forces on the downstream side, F2 and F3 A component of the weight of water in the control volume in the downstream direction, Wx Shear force, Ff , acting on the channel bottom and the sides. z = depth of the centroid of flow area A below the water surface γ = specific weight of water. The component of the weight of water in the downstream direction, θ = Angle between the channel bottom and the horizontal axis. Since the channel-bottom slope is assumed to be small, sinθ ≈ tanθ ≈ −dz/dx. Negative sign is due to the fact that z decreases as x increases. The pressure force acting on the downstream side of the control volume may be divided into two parts. F2 is the pressure force due to flow depth, y F3 is the pressure force for the increase in depth in distance, Δx.  Neglecting the higher-order term, which corresponds to the small triangle at the top. If the average shear stress acting on the channel bottom and sides is τo, Shearing force P = wetted perimeter Resultant force, Fr acting on the control volume in the downstream direction, (1) To apply the Reynolds transport theorem, the intensive property, β = V. (2) Since the flow is assumed to be steady, the first term on the right-hand side of this equation is zero. From (1) and (2) R = A/P = hydraulic radius (3) Sf = slope of the energy grade line = −dH/dx. Negative sign since H decreases as x increases.  If the flow is steady and uniform, dV/dx = 0 and dy/dx= 0. (4) Based on a dimensional analysis, (5) k is a dimensionless constant that depends upon the Reynolds number, roughness of the channel bottom and sides, etc. From (3) and (5) C = Chezy constant valid for non-uniform, steady flow This equation was introduced by a French engineer named Antoine Chezy, in 1768 while designing a canal for the water supply system of Paris.  For uniform flow From (4) and (5) Sf is slope of the energy grade line for non-uniform flow. So is slope of the channel bottom (which has the same value as the slope of the energy grade line or the slope of the water surface) for the uniform flow. C has dimensions of √length/time as compared to the Darcy Weisbach friction factor, f which is dimensionless. Like f, C depends upon the channel roughness and the Reynolds number, Re. In addition, it may depend upon the channel cross sectional shape, this dependence appears to be small [Anonymous, 1963] and may be neglected. Because the channel roughness may vary over a wide range, its effect on C has not been as thoroughly investigated as that on f.  Darcy-Weisbach friction formula for pipes hf = head loss in a pipe of diameter D and length L. The slope of the energy grade line, S = hf/L. Hydraulic radius, R, for a pipe is equal to D/4, Moody diagram plotted with C as the ordinate instead of f [Henderson, 1966] This diagram is divided into three regions: hydraulically smooth, transition, and fully rough.  A flow may be considered hydraulically smooth even though the channel surface is rough provided the projections of the surface roughness are covered by the laminar sublayer. As reynolds number increases, the thickness of this layer decreases and the effect of roughness projections on flow becomes important. Then, the flow is in the transition region. When the roughness projections are not covered by the viscous sub- layer and dominate the flow because losses are due to form drag, flow may be classified as fully rough. Flow types may be classified based on the value of a dimensionless number, Rs = kV∗/ν. ν is the kinematic viscosity of the liquid; k is a characteristic length parameter for the size of the channel-surface roughness and V∗ is the shear velocity.  The flow is considered Rs = kV∗/ν. smooth if Rs < 4; transition if 4 < Rs < 100; fully rough if Rs > 100. The expressions for C for smooth and rough flows derived from the experimental data on flow through pipes are [Henderson, 1966]: Smooth flows Rough flows valid only for small channels with fairly smooth surfaces since these are based on pipe data. Empirical relationships and field observations should be employed for large channels with rough flow surfaces. Several forms of expressions for the Chezy coefficient C G. K. (Ganguillet and Kutter, Swiss engineers ) Formula (1859) Expressed the value of C in terms of the slope, hydraulic radius R, and the coefficient of roughness n. In English units, In MKS units  The Bazin Formula Proposed by the French hydraulician H. Basin in 1897 Chezy's C is considered a function of R but not of S. In English units m is coefficient of roughness  The Powell Formula In 1950, Powell suggested a logarithmic formula for the roughness of artificial channels. R is the Reynolds number, R is the hydraulic radius in ft and є is a measure of the channel roughness For rough channels, the flow is generally so turbulent that R becomes very large compare with C. For smooth channels, the surface roughness may be so slight that є becomes negligible compared with R.  Pavlovski formula (proposed in 1925) In Metric units The exponent x depends on the roughness coefficient and hydraulic radius. The formula is valid for R between 0.1 and 3.0 m and for n between 0.011 and 0.040.  The Manning Formula First proposed by the Irish engineer Robert Manning in 1889 In English units V is the mean velocity in fps, R is the hydraulic radius in ft, S is the slope of energy line, and n is the coefficient of roughness, specifically known as Manning‘s n. In MKS units Relationship between Chezy's C and Manning's n  Factors affecting Manning's Roughness Coefficient Surface roughness – Size and shape of the grains of the material forming the wetted perimeter and producing a retarding effect on the flow. – Fine grains result in a relatively low value of n and coarse grains, in a high value of n. Vegetation – Reduces the capacity of the channel and retards the flow. Channel Irregularity – Gradual and uniform change in cross section, size, and shape will not appreciably affect the value of n, but abrupt changed or alternation of small and large sections necessitates the use of a large value of n.  Channel Alignment – Smooth curvature with large radius will give a relatively low value of n, whereas sharp curvature with sever meandering will increase n. Silting and Scouring – Silting may change a very irregular channel into a comparatively uniform one and decrease n whereas scouring may do the reverse and increase n. Obstruction – Presence of log jams, bridge piers, and the like tends to increase n. Size and Shape of Channel – An increase in hydraulic radius may either increase or decrease n depending on the condition of the channel. Stage and Discharge – The n value in most streams decreases with increase in stage and in discharge. – However, the n value may be large at high if the banks are rough and grassy.  Seasonal Change – Owing to the seasonal growth of aquatic plants, grass, weeds, willow, and trees in the channel or on the banks, the value of n may increase in the growing season and diminish in the dormant season. Suspended Material and Bed Load – The suspended material and the bed load, whether moving or not moving, would consume energy and cause head loss or increase the apparent channel roughness. where n0 is a basic n value for a straight, uniform, smooth channel in the natural material involved, n1 is a value added to n0 to correct for the effect of surface irregularities, n2 is a value for variations in shape and size of the channel cross section, n3 is a value for obstructions, n4 is a value for vegetation and flow conditions, and ms is a correction factor for meandering of channel.  COMPUTATION OF UNIFORM FLOW The Manning’s formula for uniform flow in terms of discharge Co = 1.49 in English units and Co = 1 in SI units. For a given channel section and specified bottom slope, only one discharge is possible for a given normal depth. K is called the conveyance of the channel and expresses the discharge capacity of the channel per unit longitudinal slope. K is a function of the normal depth, properties of the channel section and Manning n. The left-hand side is referred to as the section factor For the specified values of n, Q, and So, we solve this equation to determine the normal depth in a given channel. This may be done by Design Curves Trial-and-error procedure Numerical methods for the solution of a nonlinear algebraic equation Design Curves  Compute the normal depth in a trapezoidal channel having a bottom-width of 10 m, side slopes of 2H to 1V and carrying a flow of 30 m3/s. The slope of the channel bottom is 0.001 and n = 0.013. Design curves  From Design curve Trial-and-Error Procedure DESIGN OF CHANNELS FOR UNIFORM FLOW Nonerodible Channel – Most lined channels are built-up channels can withstand erosion satisfactorily and are therefore· considered nonerodible. – Unlined channels are generally erodible, except those excavated in firm foundations, such as rock bed. – The factors to be considered in the design are The kind of material forming the channel body, which determines the roughness coefficient, The minimum permissible velocity to avoid deposition if the water carries silt or debris, The channel bottom slope and side slopes, The freeboard, The most efficient section, either hydraulically or empirically determined.  The minimum permissible velocity (nonsilting velocity) – The lowest velocity that will not start sedimentation and induce the growth of aquatic plant and moss. – A mean velocity of 2 to 3 fps may be used safely when the percentage of silt present in the channel is small. – A mean velocity of not less than 2.5 fps will prevent a growth of vegetation that would seriously decrease the carrying capacity of the channel.  Nonerodible material and lining – Include concrete, stone masonry, steel, cast iron, timber, glass, plastic, etc. – The selection of the material depends mainly on the availability and cost of the material, the method of construction, and the purpose for which the channel is to be used. – The purpose of lining a channel is in most cases to prevent erosion, but occasionally it may be to check seepage losses. – In lined channels, the maximum permissible velocity, i.e. the maximum that will not cause erosion, can be ignored; provided that the water does not carry sand, gravel, or stones.  Channel Slopes – The longitudinal bottom slope of a channel is generally governed by the topography and the energy head required for flow of water. – In many cases the slope may depend also upon the purpose of the channel. – For example: channels used for water distribution purpose such as those used in irrigation, water supply, hydraulic mining, and hydropower projects require high level at the point of delivery, therefore small slope is desirable in order to keep loss in elevation to minimum. – The side slope of a channel depends mainly upon the kind of material.  Freeboard – The vertical distance from the top of the channel to the water surface at the design condition. – Should be sufficient to prevent waves or fluctuations in water surface from overflowing the sides. – Free boards varying from less than 5 % to greater than 30 % of the depth of flow are commonly used in design. – Bureau recommends that preliminary estimates of the freeboard required under ordinary conditions: F is the freeboard in ft y is the depth of water in the canal in ft C is a coefficient varying from 1.5 for a canal capacity of 20 cfs to 2.5 for a canal capacity of 3,000 cfs or more. Recommended freeboard and height of bank of lined channels. (U.S. Bureau of Reclamation)  The best hydraulic section – The conveyance of a channel section increases with increase in the hydraulic radius or with decrease in the wetted perimeter. – The channel section having the least wetted perimeter for a given area has the maximum conveyance such a section is known as the best hydraulic section. – The principle of the best hydraulic section applies only to the design of nonerodible channels. For erodible channels, the principle of tractive force must be used to determine an efficient section. BEST HYDRAULIC SECTIONS  Show that the best hydraulic trapezoidal section is one-half of a hexagon. Water area and wetted perimeter of a trapezoid y is the depth, b is the bottom width, and z:1 is the side slope Consider A and z to be constant For a minimum wetted perimeter, dP/dy = 0 Substituting this equation for b in the previous two equations for A and P Find the value of z that makes P the least. Equating dP/dz to zero, and solving for z This means that the section is a half hexagon.  A trapezoidal channel carrying 400 cfs is built with nonerodible bed having a slope of 0.0016 and n = 0.025. Proportion the section dimensions. Assuming b=20 ft and z = 2 Assign a freeboard of 2 ft. Total depth of the channel is 5.36 ft. The top width of the channel (not the width of the water surface) is 41.4 ft. The water area is 89.8 ft2, and the velocity is 4.46 fps, which is greater than the minimum permissible velocity for inducing silt, if any. AR2/3= 167.7 y = 6.6 ft. Add 3 ft freeboard, the total depth is 9.1 ft. The corresponding bottom width is 7.6ft. The top width of the channel is 18.7 ft. The water area is 75.2 ft2. The velocity is 5.32 fps.