#### Transcript Solution of the St Venant Equations / Shallow

Solution of the St Venant Equations / Shallow-Water equations of open channel flow Dr Andrew Sleigh School of Civil Engineering University of Leeds, UK www.efm.leeds.ac.uk/CIVE/UChile Background information Why should we model rivers? It is difficult and expensive to get data The flow changes from day to day Most of the time they are no problem They cause disruption They are dangerous They Cause Financial and Personal loss They cause structural damage Human interference does not help They are not new Preventative Measures • • • • build higher flood banks divert the water with a relief channel store the water a combination of these Design Considerations Appearance Effects on both upstream and downstream The cost The flood return period Data availability Consider that … Floods cannot be prevented It is neither economic nor practical to design for exceptional floods The Elements of Flood Hydraulics Flood routing is the process of calculating backwater curves in unsteady flow. Why do we need to route floods? To know: Extent of flooding Effects hydraulic structures e.g. bridge piers, culverts, weirs Size of flood relief channels If flood relief measures will work Give flood warnings For each return period Take the flood hydrograph Route this flow through the system Ask if your proposal will work Repeat for every proposal and return period Objectives of this course Understand necessary computational components See different form of equations of unsteady flow Use appropriate solution techniques By the end will have programmed a model capable of simulating passage of a flood wave on a simple river network have programmed a model to simulate extreme open channel flows and tested this with a dam break But Today – just steady flow – like HEC-RAS steady Functions / Programs We will develop programs Matlab functions equations Graphical representation (could be any program / language) 1-D and 2-D Input data Solution data Steady / Time dependent Put function together for complete model 2-d : Layout of Network Section / Solution Profile / Solution 3-d, gis? Flood routing achieved using the St. Venant Equations d d u g u g S o S f x x t A u d u A b 0 x x t St Venant Assumptions of 1-D Flow Flow is one-dimensional i.e. the velocity is uniform over the cross section and the water level across the section is horizontal. The streamline curvature is small and vertical accelerations are negligible, hence pressure is hydrostatic. The effects of boundary friction and turbulence can be accounted for through simple resistance laws analogous to those for steady flow. The average channel bed slope is small so that the cosine of the angle it makes with the horizontal is approximately 1. Dam Break: real and dangerous Dam break: difficult to solve Idealised case Sharp gradients Dam Break: Animation By the end of the course will be able to do something like this. Basics: Consider Steady Flow Today’s class will cover: Components of a computational model How to represent a network Fundamental (steady) equations Section properties Friction formulas Conveyance Steady solutions: uniform flow, backwater curve. How to represent channel network Sections Reach – group of sections Boundary conditions Internal: join reaches “External”: define inflow and outflow Together define river system Diagrammatic picture Sections Look downstream. Left bank, Right bank Sections Variable roughness, shape, across section Sections: File Format Local coordinates: x along channel, y across, z vertical SECTION 8 0 5 15 45 47.5 60 65 75 AV2296_11909 22.61 19.89 14.44 14.44 17 17 18.87 22.61 0.5 0.04 0.04 0.04 0.5 0.5 0.5 0.5 Section Properties Depth (d or y) – the vertical distance from the lowest point of the channel section to the free surface. Stage (z) – the vertical distance from the free surface to a datum Area (A) – the cross-sectional area of flow, normal to the direction of flow Wetted perimeter (P) – the length of the wetted surface measured normal to the direction of flow. Surface width (B) – width of the channel section at the free surface Hydraulic radius (R) –area to wetted perimeter ratio (A/P) Hydraulic mean depth (Dm) –area to surface width ratio (A/B) Hydraulic diameter (DH) = equivalent pipe diameter (4×R = 4A/P = D for a circular pipe flowing full) Centre of gravity coordinates (centroid) Function for Section Properties Any section defined by coordinates (in file) Common sections: y 1 y x 1 8 sin D 2 b 2 y 1 x2 1 2 D b+2xy sin 2D Area, A by (b+xy)y Wetted perimeter P b 2y Top width B B Hydraulic radius R by b 2 y b xy y b 2y 1 x depth Dm D b b mean Y B B B Hydraulic Circle Trapezoid Rectangle b xy y b 2 xy 2 1 sin D 1 4 1 sin D 8 sin 1 / 2 d Steady Flow Equations Conservation of energy Energy / Bernoulli Equation p1 V12 p2 V22 1 z1 2 z2 H constant g 2g g 2g p hydrostatic pressure distribution y d cos g Bed slope small: tan θ ≈ sin θ ≈ θ in radians H V2 1 V2 f z d cos S f s s 2g R 2g Momentum Equation When flow is not hydrostatic, steep, discontinuous etc. Hydraulic Jump F Q 2V2 1V1 =momentum correction factor Velocity Distribution Velocity Distribution on Bend Hitoshi Sugiyama. See animation. http://www.cc.utsunomiya-u.ac.jp/~sugiyama/avs4/avs4eng.html Calculation of and 3 u dA V13 A1 V33 A2 V33 A3 3 V A V 3 A1 A2 A3 V Q/ A 2 u dA V12 A1 V32 A2 V32 A3 2 V A V 2 A1 A2 A3 Q V1 A1 V2 A2 V3 A3 V A A1 A2 A3 Function: Calculate the coefficients α and β for a given section and vel dist. Reynolds Numebr Using R as length scale uR uD Re pipe Re R channel 4 4 Using DH as length scale Re DH channel uDH uD Re pipe For a wide river R = depth, DH = 4×depth. Function: Calculate Re (ReR or ReDH) for a given fluid, section, depth and velocity. Froude Number, Fr V Fr gd A g B Critical Depth Fr = 1 Fr < 1 sub-critical Fr V upstream levels affected by downstream controls Fr > 1 super-critical upstream levels not affected by downstream controls Function: Calculate Fr, for a given section and discharge. Also dcritical. Uniform Flow Equilibrium – Friction balances Gravity o So sin gR H z So s s So S f L V z weight s bed x Function: Calculate bed shear stress, o for given section, depth and bed slope. Chezy C assuming rough turbulent flow shear force is proportional to velocity squared o V 2 o kV 2 thus V g k RSo V C RSo Functions:Calculate V or Q for a given section and dn, C and bed slope. Also: normal depth, dn from Q, C, So, C from Q and So, dn, So from C, Q, dn. Friction Formulae Darcy-Weisbach for pipe Full pipe S o = L / hf and 4 fLV 2 hf 2 gD R A/ P D / 4 1 V2 So f R 2g C 2g f 2 gRS o f V2 Alternative form for f Some texts give the value f is 4 times larger than quoted here To clarify some text use l such that: hf lLV 2 2 gD l 8 gRS o V2 BE CAREFUL WITH FRICTION FORMULAE Functions: Calculate f or λ for a given section, depth, slope and discharge. Calculate f from C and vice versa Colebrook-White equation for f Originally developed for pipes ks 1 1.26 4 log10 f 14.8R Re R f ks is effective sand grain size in mm Implicit Requires iterative solution Use Altsul equation to start iteration 1/ 4 k 25 f 0.4 0.365 s R Re R ks values Some typical values of ks are Material ks (mm) Plastic 0.01 – 0.02 Painted pipe 0.02 Riveted steel 1-10 Cast iron (new) 0.25 Cast iron (rusted) 1-1.5 Concrete 0.3-3 Concrete (rough) 3-10 Planed wood 0.6-2 Rubble 5-10 Straight earth channel 3 Function: Calculate f or λ from ReR depth, section and ks. Manning’s n Most commonly used expression for friction R 2 / 3 S o1/ 2 V n n relates to C: R1 / 6 C n In terms of discharge 1 A5 / 3 1 / 2 Q S 2/3 o nP Function: Calculate Q from n, C from n, for given section. Manning’s n values Some typical values for n Channel type Surface material and form Manning’s n range River earth, straight 0.02-0.025 earth, meandering 0.03-0.05 gravel (75-150mm), straight 0.03-0.04 gravel (75-150mm), winding 0.04-0.08 Grass / light growth 0.05 Trees 0.15 earth, straight 0.018-0.025 rock, straight 0.025-0.045 lined canal concrete 0.012-0.017 lab. models mortar 0.011-0.013 Perspex 0.009 Flood plain unlined canal Friction estimate great source of error Computations in uniform flow Typical and common calculations Discharge from a depth = normal flow Depth for a discharge = normal depth Require iterative solution even for rectangular channel Function: Calculate dn or flow for given section and n, C or f , So, Q or dn. Conveyance, K K measure of carrying capacity of a channel in uniform flow Chezy: Q AC RSo K So K ACR1/ 2 Manning A5 / 3 1 K 2 / 3 AR2 / 3 nP n Function: Calculate conveyance for a given section and n, C or f. Conveyance in Irregular Channels Split section into regions of “uniform” velocity Separate flood plain and main channel. Regions could be defined by roughness Function: Calculate conveyance for irregular section must define a subdivision method Calculate α for irregular channel with sub division by specified roughness Exercises Calculations Uniform flow exercise questions ExerciseQuestions02.pdf on web page Questions: 1-7 Backwater Calculation Gradually varied flow surface profile Calculated from Energy / Bernoulli equation Basis of HEC-RAS Steady Backwater calculations are developed assuming: Non-uniform flow Steady flow Flow is gradually varied That at any point flow resistance is the same as for uniform flow i.e can use manning of Chezy etc. Backwater Calculation 2 Start at known depth and Q, integrate up or down stream Control section: Critical depth, change in slope, structure, hydraulic jump Super-critical at control section: forward integration (downstream) Sub-critical at control section: backwards integration (upstream). Backwater finite difference e.g. energy equation with Manning H V2 n 2V 2 4 / 3 z d cos s s 2g R Vi2/2g total head 2/2g Vi+1 yi di free surface y i+1 zi d i+1 s zi+1 x bed H i1 H i 0.5 S fi 1 S fi si1 si Backwater Calculation Procedure 1. 2. 3. 4. 5. At point of known depth and Q, si. Calculate Ai, Pi, Vi, Sf_i Hi, Estimate di+1, calculate properties at i+1, H(1)i+1 Calculate H(*)i+1 using FD form of energy equation If H(1)i+1 not close to H(*)i+1 (e.g. 1mm) repeat from step 2. Else carry on integration further along channel Functions: Integrate backwater for a prismatic channel.. Also a similar function for a channel defined by a series of cross sections. Backwater Exercise Backwater integration exercise questions ExerciseQuestions02.pdf Question: 8 Should be straight forward using developed functions.