Solution of the St Venant Equations / Shallow-Water equations of open channel flow Dr Andrew Sleigh School of Civil Engineering University of Leeds, UK

St. Venant Equations

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. Cunge J A : Practical Aspect of Computational River Hydraulics

Control Volume

Continuity CV in x,t plane  between cross sections x=x1 and x=x2  between times t=t1 and t=t2 conservation of mass

Momentum Conservation of momentum

Geometric Change Terms vertical change in cross-section change in width along the length of the channel. dd h-   b h ()()

Integral / Differential Forms Integral form do not require that any flow variable is continuous We will see later finite difference methods based on this integral form. Can derive differential form … but  Must assume variables are continuous and diferentable  Replace variable with Taylor’s series

Differential form -

In terms of Q(x,t) and h(x,t): Where b = b(h), A=A(h) remember b = b(h), A=A(h) Each of these forms are a set of non-linear differential equations which do not have any analytical solution. The only way to solve them is by numerical integration.

In term of u(x,t) and h(x,t): using

Characteristic Form The St Venant equations may be written in a quite different form know as the Charateristic Form. Writing the equations in this form enables some properties and behaviour of the St Venant equations become clearer. It will also help identify some stability criteria for numerical integration will help with the definition of boundary conditions.

Characteristic form Consider a prismatic channel

Wave speed Consider the speed, c, of a wave travelling in the fluid.  with respect to x and t gives:

Combining Adding equations (3) and (4) gives Subtracting equations (3) and (4) gives

Characteristics Equations (5) and (6) can be rearranged to give respectively

Total differential For some function  of x and t In a physical sense.  If the variable  is some property of the flow e.g. surface level,  if an observer is moving at velocity v,  the observer will see the surface level change only with time relative to the observers' position.

The characteristic form of the St Venant equations If we take  = (v + 2c)  Total differential is  Compare with  Clearly  and

Characteristic form of the St Venant Equations These pairs are known as the Characteristic form of the St Venant Equations for

Meaning of the characteristics The paths of these observers that we have talked about can be represented by lines on this graph.

Information paths it takes time for information to travel E.g. a flood wave at u/s end The channel downstream will not receive the flood for some time. For how long? The line C0 represents the velocity of flood wave Everything below C0 is zone of quite

Zones of Dependence and Influence The idea that characteristics carry information at a certain speed gives two important concepts

Stability These zones imply a significance to numerical methods and stability of any solver The numerical method must take only information from within the domain of dependence of P this limits the size of time step

Calculation with characteristics If we know the solution at points 3 and 5 Can determine the solution at point 6.  For C5-6 then  For C3-6 then

Characteristics solution Adding equations Subtracting

MOC on Rectangular Grid

Midstream discretisation Away from boundaries

Solution

Stability Considering characteristics

Boundary conditions Upstream boundary:  backward characteristic Downstream boundary:  forward characteristic

Boundary Conditions The second equation is a  boundary condition equation Upstream depth boundary hP = H(t)

Spuer-critical - mid Right characteristic moves to left solution method is exactly the same

Super-critical - upstream No characteristics

Super-critical downstream 2 characteristics  No boundary condition

Finite Difference Schemes Two basics classes  Implicit  Explicit Commercial packages use implicit Explict  for high accuracy (sometimes!)  Testing / understanding behaviour  Class examples!

Which Equations? Not always clear what equations a being used! What are the shallow water equation? We will look at schemes based on the integral equations:

Homogeneous Integral Equations Without the gravity / frictions terms

Grid based Consider the grid …

Integrate around the cell Considering the cell ABCD,  Integral can be written in the general vector form :

Gridpoints / Variables Variables are all known or will be calculated at the grid points  x i represents the value of x at position i  t n represents the value of t at position n Derivation approximates values:

Substitute in approximations And the equation become

Difference equation Divide through byΔx Δt And can see that it is an approximation of i.e. starting with integral form discretisation is also valid for diferential form

Several schemes This is a general discretisation scheme  Vary the parameters ψ and θ  Get a family of different finite difference schemes Features are  They are implicit for values of ψ > 0. else explicit.  They link together only adjacent nodes. Space interval can vary – no loss of accuracy.  They are first order, except for the special case of ψ=θ=0.5 when they are second order.

Preissmann Scheme ψ = 0.5 gives Preissmann 4-point scheme Time derivative Space derivative

Equations become …

More common form The terms I1 and I 2 are often difficult (expensive / time consuming)  (when originally attempted) Usual form used in packages

Priessmann Function represented by, 0.5 < θ ≤ 1 Continuity :

Priessmann Momentum Equation

Source term So-Sf

Unknowns There are the 4 unknowns Plus k, b, A which are functions of h and Q

Need to linearise “Linearise” equations A, B, C, D and RHS are funtion of the unknowns.  But not strongly

Boundary Conditions In implicit scheme specify h or Q  Use characteristics to decide appropriately Or relation between h and Q 2N unknowns (h, Q at each node)  2N – 2 equations from internal points  2 boundary equations

Junction At a junction each chanel share the same node  h juntion 1 = h juntion 2 = h juntion 3 …..= h juntion n = h Continuity  Sum of inflow and outflow equal to zero

Iterative solution Need to iterate updating coeficients Cunge says …  iterates rapidly one or two iterations Newton-Raphson methods used in packages

Stability Formally unconditionally stable for all time steps  0.5 < θ ≤ 1 Further away from Cr = 1 less accurate Cr = 20 is common Because of linearisation may fail for extreme flows or those that are too far from original assumptions

Explicit Schemes Not used in simulations of real rivers Time step limitations. Important features of explicit schemes  they are simple to implement  allow experiment with weights, time-step and space-step to understand behaviour of the solution. There are MANY schemes

Leap-Frog Earliest scheme aplied to wave equations  SpatialderivativeTemporal derivative gives

Stability All explict scheme have time step limit Courant confition (CFL) Cr < 1

Lax Explict Scheme Similar to leap from with weighting, α, in time For 0 ≤ α ≤ 1 Spatial derivative

Lax Explict Scheme Leads to function solution Boundary conditions must be applied using method of characteristics

Some useful texts Rather old- still basis of many commercial programs. Cunge, J.A., Holley, F.M. and Verwey, A. (1980): Practical Aspects of Computational River Hydraulics Mahmood and Yevjevich (1975): Unsteady Flow in Open Channels - Fort Collins, Colorado Liggett & Cunge (1975) Preissmann, A. (1960): Propogation des Intumescenes dans les Canaue et Rivieres - 1st Congress de l'Assoc. Francaise de Calcul, Grenoble