The University cannot take responsibility for any misprints or errors in the presented formulas. Please use them carefully and wisely.

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Ethelbert Lloyd
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1 Aide Mémoire Suject: Useful formulas for flow in rivers and channels The University cannot take responsiility for any misprints or errors in the presented formulas. Please use them carefully and wisely. 1. General asic equations The general continuity equation for flow, following from conservation of mass, is: + div( v r ) 0 For an incompressile fluid D/Dt0. Furthermore for a homogeneous fluid also grad()0, leading to the following continuity equation (conservation of volume): divv r 0 From the asic equations for conservation of momentum the commonly used equation of motion of Navier-Stokes can e derived (assuming homogeneous incompressile fluid with constant density ): r u r r 1 r r 1 + ( u grad) u + grad p + grad( gz) ν u K local acceleration pressure gravity internal external and convection friction forces The equation of Euler can e derived y neglecting of internal friction terms (viscous terms) The relation etween convective term and viscous shear-stress term can e expressed in u / L ul terms of a Reynolds numer: Re νu/ L ν - Small Re: stale system, laminar flow, Navier-Stokes is linear. - Large Re (> O(10 )): non-linear convection dominates, system is instale (small changes give large differences in solution), turulent flow. The internal friction terms or viscous terms are diffusive terms. The external forces are mainly the shear stresses on the ed, walls and surface (wind-shear). The sudivision of units to time-average and erratic components (e.g., u +u ) will lead to alance equation for the average components that is called the Reynold equation. It is commonly used for turulent flows. Page 1

2 . Equations for Two dimensional flow Basic equations The asic equations for DH flow follow from depth integration of the Reynolds equations. The continuity equation for flow is: a au av The asic equations of motion (in their commonly used form): u + u u + v u + g v + u v + v v + g in which ( a + z ) τ ( at ) ( at x 1 xx 1 xy ) + + a ( a + z ) τ y 1 ( atxy ) 1 ( atyy ) + + a u depth-average velocity in x-direction v depth-average velocity in y-direction a local water depth z local ed level τ x, τ y ed-shear stress in x,y-direction T xx, T xy, T yy horizontal exchange of momentum through viscosity, turulence, spiral flow, and non-uniformity of velocity distriution Shear stresses can e expressed y D Chézy relations: τ x gu u C + v and τ y gv u C + v with C Chézy coefficient Spiral flow The D effect of the D spiral flow pattern due to curvature of the streamlines can e otained from a parametrised model. The effect of spiral flow on the direction of ottom-shear stress can e otained from: v δ arctan arctan A a u R* regular flow spiral flow where δ is the angle with x-coordinate axis, R * is the effective radius of the streamline, and A is the spiral flow coefficient expressed as: ε g A 1 κ κc with ε is a caliration coefficient, and κ is the Von Karman coefficient ( 0.4) Page

3 In an equilirium situation (at the end of a long end) the spiral flow intensity I is expressed y a u + v I R * Relaxation For adaptation of D non-uniform horizontal velocity distriution to a D uniform equilirium distriution a length scale is defined as (relaxation length): λ w C h g. Equations for one dimensional flow Dynamic flow (Saint-Venant equations) Integration of the continuity equation yields: B a Q + q in 0 in which a is water depth, B is the total width (including the storage width), and q in is lateral inflow Integration of the equation of motion yields for conservation of momentum: Q Q τ α ga z w + + s As + qinu A in which A s is cross-sectional area of flow, u is the x-component of velocity of lateral inflow, z w is the water surface elevation, τ is the ed-shear stress (hydraulic roughness). Comination of the equations for motion and continuity aove yields the following variant for the equation of motion: u τ u u g z w qin + + A u u s ( ) local acceleration pressure roughness lateral and convection and gravity inflow guu gqq Hydraulics roughness is normally expressed y the Chézy formula: τ in C R C As R which R Hydraulic radius (see uniform flow). (Alternative roughness formulations: see elow) In a river-flood wave the dynamic wave approach can e used that assumes that the intertia term (local acceleration term) can e neglected. The flood wave is then expressed y the following diffusion model (after simplification for A s B s u, Ra, constant C and B) : a Q a C Bs a + Ba QB a 0 Page

4 with B is the total width(including the storage width). From the general diffusion model follows the propagation speed of the wave c: 1 Q Q c B a Ba ub B s Damping of the flood wave is governed y the diffusion coefficient, which is expressed as: { 1 } Q K Fr with Bs / B Bi Damping of the flood wave occurs proportional with [ (πkt)] -1 A more simplified approach than the dynamic-wave approach can e used for relatively flat flood waves. By assuming that the flow (surface slope) ehaves like a uniform flow, the kinematic wave approach is otained. Different than the dynamic wave, the kinematic wave does not Uniform flow (Chézy formula) The uniform-flow equations follow from the Chézy formula u C Ri with R AP with i local hydraulic gradient or energy gradient, and P wetted perimeter of the cross section with flow area A. For a rectangular channel with free surface and B>>a: R pipe RD/4 with D is the diameter of the pipe. Ba a, while for a filled circular B + h For the equilirium depth follows: q ae C i The Froude numer Fr, representing the alance etween inertia and gravity, and characteristic for sudivision of su-critical (Fr<1) and supercritical flow (Fr>1, torrential regime), is defined as: Fr u ga For Fr 1 the flow is critical. Then the depth equals the critical depth a g expressed as: a q g g Hydraulic roughness Hydraulic roughness is expressed in terms of the Chézy (C), Manning-Strickler (n), Darcy- Weisach (f). The relation etween C and f is: Page 4

5 fc 8g The coefficients are generally written as C, g 1R k + /,5 18 1R log log κ δ k + δ /,5 and f 0,4 - log 1R k + δ /,5 In these equations k is the equivalent sandroughness according to Nikuradse. For an alluvial ed the value of k varies strongly with the flow conditions. In rivers the flow regime will often e hydraulically rough (k >> δ). The value of C then ecomes according to White-Colerook: 1R White - Colerook: C 18log k or according to Strickler: Strickler: C 5 R k 1/6 Most often used, and linked with Strickler's equation, is the Manning roughness formula (or Manning-Strickler roiughness formula). The relation etween Manning's roughness coefficient n and the Chézy coefficient C is (with R in meters): C 16 / R n Page 5

6 Channels Minor natural streams (Width at flood < 0 m) Lined channels Excavated dredged Natural stream on plain Mountain streams Flood plains Major stream Ven Te Chow (1959) Manning n-value Smooth metal Corrugated metal Wood Concrete Brick Asphalt Straight earth, clean Straight, gravel Rock-cut Clean.straight, full stage, no pools Idem, more stones and weeds Clean, winding, some pools and shoals Idem, more stones and weeds Idem, lower stages. More ineffective slopes and sections Sluggish reaches, weedy, deep pools Very weedy reach, deep pools, floodways with timer and underrush Vegetated steep anks, gravel, coles and few oulders Vegetated steep anks, coles with large oulders Pasture, no rush, grass Cultivated areas Scattered rush, heavy weeds Light rush and trees Medium to dense rush dense willows, summer, straight cleared land with tree stumps and sprouts heavy stand of timer, little undergrowth Regular section, no oulders of rush Irregular and rough section Tale Range of values of the roughness coefficient n for different types of channels Surface profiles Page 6

7 The equation of Bélanger holds for a channel with constant width and ed slope i (i.e., following from one-dimensional-flow asic equations or Saint-Venant equations simplified for stationary flow): d h d x i e g h - h h - h h h and 1- h / h 1- Fr follows: With η e β { g e} x h i e η-β d η + 1- η constante to e solved using the Bresse method: ψ( η) dη 1 - η 1 6 ln η + η + 1 ( η - 1 ) 1 η arctan - arctan 1 After introduction of a dimensionless length scale Λ x i h e follows from equations aove: Λ { } { η η} βψη ( ) ψ ( η ) - Λ If at x 0 holds that h 0 η 0 h e then the Bresse method can e used to determine (if Fr << 1, hence β 1) for which x still half (50%) of the induced ackwater effect remains. By fitting of Bresse function through a power function the following approximation can e otained: 4 / Λ 1 / 0, 4η 0 For relatively small ackwater effects (η 0 1) the value Λ 1/ 1/4 is otained. These approximations are useful for sketching surface profiles. Flow over weirs, spillways and other structures Flow over a road-crested weir can e sumerged (drowned flow) or free (overflow or modular Page 7

8 flow). The discharge through the weir q can e expressed as function of the upstream velocity head H u (energy level) and the water depth downstream h d, oth relative to the crest level. Broad-crested weir, sumerged flow q mh g(h - h) with: 09, < m < 1, ( average m 11,) Broad-crested weir, free flow q m g H / These formulas are used for sharp-crested weirs as well, ut usually with a higher discharge coefficient (m). Weirs with olique flow: If in a stream channel with discharge q 0 flow passes a weir with an angle α, then q 0 follows from -1 q 0 q.sin α in which q follows from the regular discharge formula (for weirs in perpendicular flow) Page 8

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