Examples of free surface applications include:
of tanks e.g. petrol tankers where the petrol does not completely fill the
of chambers e.g. a barrel being filled with oil
emptying e.g. a toilet cistern flushing a toilet pan
and jetting e.g. a water jet hosing a car
impacting static bodies e.g. a sea wall or the legs of an oil platform in
engineering hydraulics e.g. dams, weirs, orifices, valves, sluices
engineering e.g. partially filled valves, orifices, pipes
How are Free Surface Simulations Calculated
The underlying technology is called the ‘Volume of Fluid’
(VOF) method. In the case of air and water, this assigns a volume fraction of
air and water to each cell in the domain. The sum of the two fractions is
always 1, so calculating one fraction always implies the other. Where the
volume fraction is 0.5, there will be a free surface. Either side of this cell
the volume fraction will tend towards fully water or fully air. As with all
Flow simulations, the software calculates the mass and volume of fluids
entering and leaving each cell and conserves momentum, energy and mass. However,
the actual movement of the free surface is obtained using ‘transport equations’
driven by fluid momentum calculations and external boundary conditions including
gravity. More details are available in the Flow Simulation Technical Reference.
Testing the Technology
For those of us who struggle with understanding second order
differential equations (most of us!) the proof of the pudding is in the eating,
or in terms of Flow, the proof is whether the software actually works. There
are plenty of fancy animations online that look impressive but I like to be
sure of the technology before I recommend it to a customer – therefore I set
myself a free surface challenge - based on a ‘Sharp Crested Weir’.
A Sharp Crested Weir is a well understood hydraulics problem
which is taught in universities to civil engineers who need to design dams, splash
pools, water courses and weirs. Below is an extract from one university lecture
that is publicly available on YouTube.
The critical equation is derived from energy balances and
takes the form …
- Q = volumetric water flow rate over the weir in m3/s
g = gravity in m/s2
- L = the width of the weir (perpendicular to the water flow)
- H = the ‘head’ of the weir i.e. the height of the upstream
water level measured vertically from the top of the weir in metres
P = the upstream height of the weir in metres
X = a constant for a sharp crested weir = 1.5
Cd =the ‘Discharge
Coefficient’ that has been obtained by experimental work and variables H
Creating the SOLIDWORKS Model
The first step in simulating this is to build a simple model in SOLIDWORKS. This is
for both the weir and 2 bodies to represent the water – upstream (with a head)
and downstream as a splash pool. Since we will simulate a 2D slice, this can be
done with simple extrudes ignoring any channel sides.
The dimensions used were:
- H = 400 mm
- P = 1000 mm
- L = 20 mm (i.e. the
depth of the 2D computational domain)
Plugging these values into the equation gives a predicted
flow rate, Q = 0.0095 m3/s
Flow Simulation Study Setup
The set up in Flow is quite simple …
I also enabled saving of important parameters that I want to
display using the ‘Transient Explorer’. I did this in ‘Calculation Control
Options’ and chose saving of ‘Volume Fraction of Water’ and ‘Immiscible Water’.
This enables a very quick and easy way to create animations of the chosen
parameters over time.
The mesh in Free Surface simulations is very important. As a
rough guide you need at least 5 cells across the width of each fluid stream. I
used a ‘Manual’ mesh and several local meshes.
A result at 1 sec after the water is release is shown below.
The water is spilling over the weir and has generated a wave in the splash
The graph above shows the volumetric flow rate of water over
the weir (Q). At 1 sec the flow rate is 0.0082 m3/s.
The flow rate reaches a maximum value at 1.5 seconds of 0.0088 m3/s.This is within 8% of the theoretical value. This is a great start, but is there
any reason for the small discrepancy? Is the theory correct or the simulation?
To answer that we need to understand the assumptions in the
theory. The mathematical derivation (based on energy conservation) makes the
assumption that the upstream flow is constant and sufficient to perfectly match
the flow over the weir i.e. the surface of the water is at a fixed height (H and
P are constants).
In contrast the Flow simulation is modelled as a tank with
finite capacity so the water level slowly reduces. Furthermore, you can see
from the animation below that the upstream water gently sloshes from end to end
as a result of the initial release of the water.
Clearly we could extend the tank or even introduce more
water upstream with an inlet boundary condition which would result in a higher
flow rate over the weir giving an even closer match to the theory. However, I
think it is informative to observe the sloshing effect.
This is just one of a number of Free Surface applications
but the accuracy of the method gives confidence to apply the technology to more
interesting situations like a more complex 3D weir (Cipolletti weir) with a
downstream splash tank and a complex 3D outlet channel such as this.
You can even simulate the flushing of toilets!!!
(The video shows the water flushing the pan on a half model
to make it easier to see)
Solid Solutions Management - Group Technical Director