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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

Shock Capturing Methods

- Ability to examine extreme flows
- Changes between sub / super critical
- Other techniques have trouble with trans-critical
- Steep wave front
- Front speed
- Complex Wave interactions
- Alternative shock fitting
- Good, but not as flexible

More recent

- Developed from work on Euler equations in the

aero-space where shock capturing is very

important (and funding available) - 1990s onwards
- Euler equations / Numerical schemes
- Roe, Osher, van Leer, LeVeque, Harten, Toro
- Shallow water equations
- Toro, Garcia-Navarro, Alcrudo, Zhao

Books

- E.F. Toro. Riemann Solvers and Numerical Methods

for Fluid Dynamics. Springer Verlag (2nd Ed.)

1999. - E.F. Toro. Shock-Capturing Methods for

Free-Surface Flows. Wiley (2001) - E.F. Toro. Riemann Problems and the WAF Method

for Solving Two-Dimensional Shallow Water

Equations. Philosophical Transactions of the

Royal Society of London. A33843-68 1992.

Dam break problem

- The dam break problem can be solved
- It is in fact integral to the solution technique

Conservative Equations

- As earlier, but use U for vector

I1 and I2

- Trapezoidal channel
- Base width B, Side slope SL Y/Z
- Rectangular, SL 0
- Source term

Rectangular Prismatic

- Easily extendible to 2-d
- Integrate over control volume V

2-dimensions

- In 2-d have extra term
- friction

Normal Form

- Consider the control volume V, border O

Rotation matrix

- H(U) can be expressed

Finite volume formulation

- Consider the homogeneous form
- i.e. without source terms
- And the rectangular control volume in x-t space

Finite Volume Formulation

- The volume is bounded by
- xi1/2 and xi-1/2 tn1 and tn
- The integral becomes

Finite Volume Formulation

- We define the integral averages
- And the finite volume formulation becomes

Finite Volume Formulation

- Up to now there has been no approximation
- The solution now depends on how we interpret the

integral averages - In particular the inter-cell fluxes
- Fi1/2 and F1-1/2

Finite Volume in 2-D

- The 2-d integral equation is
- H(u) is a function normal to the volume

Finite Volume in 2-D

- Using the integral average
- Where V is the volume (area) of the volume
- then

Finite Volume in 2-D

- If the nodes and sides are labelled as
- Where Fns1 is normal flux for side 1 etc.

FV 2-D Rectangular Grid

- For this grid
- Solution reduces to

Flux Calculation

- We need now to define the flux
- Many flux definitions could be used to that

satisfy the FV formulation - We will use Godunov flux (or Upwind flux)
- Uses information from the wave structure of

equations.

Godunov method

- Assume piecewise linear data states
- Means that the flux calc is solution of local

Riemann problem

Riemann Problem

- The Riemann problem is a initial value problem

defined by - Solve this to get the flux (at xi1/2)

FV solution

- We have now defined the integral averages of the

FV formulation - The solution is fully defined
- First order in space and time

Dam Break Problem

- The Riemann problem we have defined is a

generalisation of the Dam Break Problem

Dam Break Solution

- Evolution of solution
- Wave structure

Exact Solution

- Toro (1992) demonstrated an exact solution
- Considering all possible wave structures a single

non-linear algebraic equation gives solution.

Exact Solution

- Consider the local Riemann problem
- Wave structure

Possible Wave structures

- Across left and right wave h, u change v is

constant - Across shear wave v changes, h, u constant

Determine which wave

- Which wave is present is determined by the change

in data states thus - h gt hL left wave is a shock
- h hL left wave is a rarefaction
- h gt hR right wave is a shock
- h hR right wave is a rarefaction

Solution Procedure

- Construct this equation
- And solve iteratively for h (h).
- The functions may change each iteration

f(h)

- The function f(h) is defined
- And u

Iterative solution

- The function is well behaved and solution by

Newton-Raphson is fast - (2 or 3 iterations)
- One problem if negative depth calculated!
- This is a dry-bed problem.
- Check with depth positivity condition

DryBed solution

- Dry bed on one side of the Riemann problem
- Dry bed evolves
- Wave structure is different.

Dry-Bed Solution

- Solutions are explicit
- Need to identify which applies (simple to do)
- Dry bed to right
- Dry bed to left
- Dry bed evolves h 0 and u 0
- Fails depth positivity test

Shear wave

- The solution for the shear wave is straight

forward. - If vL gt 0 v vL
- Else v vR
- Can now calculate inter-cell flux from h, u

and v - For any initial conditions

Complete Solution

- The h, u and v are sufficient for the Flux
- But can use solution further to develop exact

solution at any time. - i.e. Can provide a set of benchmark solution
- Useful for testing numerical solutions.
- Choose some difficult problems and test your

numerical code again exact solution

Complete Solution

Difficult Test Problems

- Toro suggested 5 tests

Test No. hL (m) uL (m/s) hR (m) uR (m/s)

1 1.0 2.5 0.1 0.0

2 1.0 -5.0 1.0 5.0

3 1.0 0.0 0.0 0.0

4 0.0 0.0 1.0 0.0

5 0.1 -3.0 0.1 3.0

Test 1 Left critical rarefaction and right

shock Test 2 Two rarefactions and nearly dry

bed Test 3 Right dry bed problem Test 4 Left

dry bed problem Test 5 Evolution of a dry bed

Exact Solution

- Consider the local Riemann problem
- Wave structure

Returning to the Exact Solution

- We will see some other Riemann solvers that use

the wave speeds necessary for the exact solution. - Return to this to see where there come from

Possible Wave structures

- Across left and right wave h, u change v is

constant - Across shear wave v changes, h, u constant

Conditions across each wave

- Left Rarefaction wave
- Smooth change as move in x-direction
- Bounded by two (backward) characteristics
- Discontinuity at edges

Crossing the rarefaction

- We cross on a forward characteristic
- States are linked by
- or

Solution inside the left rarefaction

- The backward characteristic equation is
- For any line in the direction of the rarefaction
- Crossing this the following applies
- Solving gives
- On the t axis dx/dt 0

Right rarefaction

- Bounded by forward characteristics
- Cross it on a backward characteristic
- In rarefaction
- If Rarefaction crosses axis

Shock waves

- Two constant data states are separated by a

discontinuity or jump - Shock moving at speed Si
- Using Conservative flux for left shock

Conditions across shock

- Rankine-Hugoniot condition
- Entropy condition
- ?1,2 are equivalent to characteristics.
- They tend towards being parallel at shock

Shock analysis

- Change frame of reference, add Si
- Rankine-Hugoniot gives

Shock analysis

- Mass flux conserved
- From eqn 1
- Using
- also

Left Shock Equation

- Equating gives
- Also

Right Shock Equation

- Similar analysis gives
- Also

Complete equation

- Equating the left and right equations for u
- Which is the iterative of the function of Toro

Approximate Riemann Solvers

- No need to use exact solution
- Expensive
- Iterative
- When other equations, exact may not exist
- Many solvers
- Some more popular than others

Toro Two Rarefaction Solver

- Assume two rarefactions
- Take the left and right equations
- Solving gives
- For critical rarefaction use solution earlier

Toro Primitive Variable Solver

- Writing the equations in primitive variables
- Non conservative
- Approximate A(W) by a constant matrix
- Gives solution

Toro Two Shock Solver

- Assuming the two waves are shocks
- Use two rarefaction solver to give h0

Roes Solver

- Originally developed for Euler equations
- Approximate governing equations with
- Where is obtained by Roe averaging

Roes Solver

- Properties of matrix
- Eigen values
- Right eigen vectors
- Wave strengths
- Flux given by

HLL Solver

- Harten, Lax, van Leer
- Assume wave speed
- Construct volume
- Integrate round
- Alternative gives flux

HLL Solver

- What wave speeds to use?
- Free to choose
- One option
- For dry bed (right)
- Simple, but robust

Higher Order in Space

- Construct Riemann problem using cells further

away - Danger of oscillations
- Piecewise reconstruction
- Limiters

Limiters

- Obtain a gradient for variable in cell i, ?i
- Gradient obtained from Limiter functions
- Provide gradients at cell face
- Limiter ?i G(a,b)

Limiters

- A general limiter
- ß1 give MINMOD
- ß2 give SUPERBEE
- van Leer
- Other SUPERBEE expression
- s sign

Higher order in time

- Needs to advance half time step
- MUSCL-Hancock
- Primitive variable
- Limit variable
- Evolve the cell face values 1/2Dt
- Update as normal solving the Riemann problem

using evolved WL, WR

Alternative time advance

- Alternatively, evolve the cell values using
- Solve as normal
- Procedure is 2nd order in space and time

Boundary Conditions

- Set flux on boundary
- Directly
- Ghost cell
- Wall u, v 0. Ghost cell un1-un
- Transmissive Ghost cell hn1 hn
- un1 un

Wet / Dry Fonts

- Wet / Dry fronts are difficult
- Source of error
- Source of instability
- Common
- near tidal boundaries
- Flooding - inundation

Dry front speed

- We examined earlier dry bed problem
- Front is fast
- Faster than characteristic.
- Can cause problem with time-step / Courant

Solutions

- The most popular way is to artificially wet bed
- Give a small depth, zero velocity
- Loose a bit of mass and/or momentum
- Can drastically affect front speed
- E.g. a 1.0 dambeak with 1cm gives 38 error
- 1mm gives 25 error try it!

Conservation Errors

- The conserved variable are h and hu
- Often require u
- Need to be very careful about divide by zero
- Artificial dry-bed depths could cause this

Source Terms

- Lumped in to one term and integrated
- Attempts at upwinding source
- Current time-step
- Could use the half step value
- E.g.

Main Problem is Slope Term

- Flat still water over uneven bed starts to move.
- Problem with discretisation of

Discretisation

- Discretised momentum eqn
- For flat, still water
- Require

A solution

- Assume a datum depth, measure down
- Momentum eqn
- Flat surface

and

Some example solutions

Weighted mesh gives more detail for same number

of cells

Dam Break - CADAM Channel with 90 bend

Secondary Shocks