.. vim: syntax=rst
CHAPTER 2. FLO-2D MODEL THEORY
==================================
FLO-2D is a simple volume conservation model.
It moves the flood volume around on a series of tiles for overland flow or through stream segments for channel routing.
Floodwave progression over the flow domain is controlled by topography and resistance to flow.
Flood routing in two dimensions is accomplished through a numerical integration of the equations of motion and the conservation of fluid volume for
either a water flood or a hyperconcentrated sediment flow.
A discussion of the governing equations follows.
Governing Equations
---------------------
The constitutive fluid motion equations include the continuity equation and the momentum equation:
.. math::
:label:
\frac{\partial h}{\partial t} + \frac{\partial\, hV}{\partial x} = i
.. math::
:label: friction_slope_equation
S_f = S_o
- \frac{\partial h}{\partial x}
- \frac{V}{g}\, \frac{\partial V}{\partial x}
- \frac{1}{g}\, \frac{\partial V}{\partial t}
where h is the flow depth and V is the depth-averaged velocity in one of the eight flow directions x.
The excess rainfall intensity (i) may be nonzero on the flow surface.
The friction slope component S\ :sub:`f` is based on Manning’s equation.
The other terms include the bed slope (S\ :sub:`o`) pressure gradient and convective and local acceleration terms.
This equation represents the one-dimensional depth averaged channel flow.
For the floodplain, while FLO-2D is multi-direction flow model, the equations of motion in FLO-2D are applied by computing the average flow velocity
across a grid element boundary one direction at time.
There are eight potential flow directions, the four compass directions (north, east, south and west) and the four diagonal directions (northeast,
southeast, southwest and northwest).
Each velocity computation is one-dimensional and is solved independently of the other seven directions.
Since the flow is being shared with all of a given grid element neighbors, resolution of the velocity vectors is not required.
The stability of this explicit numerical scheme is based on strict criteria to control the magnitude of the variable computational timestep.
The equations representing hyperconcentrated sediment flow are discussed later in the manual.
Understanding, the relative magnitude of the acceleration components to the bed slope and pressure terms is important.
Henderson (1966) computed the relative magnitude of momentum equation terms for a moderately steep alluvial channel and a fast-rising hydrograph as
follows:
.. raw:: html
Bed Pressure Convective Local
Slope Gradient Acceleration Acceleration
Momentum Equation Term: So ∂h/∂x V∂V/g∂x ∂V/g∂t
Magnitude (ft/mi) 26 0.5 0.12 - 0.25 0.05
This illustrates that the application of the kinematic wave (S\ :sub:`o` = S\ :sub:`f`) on moderately steep slopes with relatively steady, uniform
flow is sufficient to model floodwave progression and the contribution of the pressure gradient and the acceleration terms can be neglected.
The addition of the pressure gradient term to create the diffusive wave equation will enhance overland flow simulation with complex topography.
The diffusive wave equation with the pressure gradient is required to predict floodwave attenuation and the change in storage on the floodplain.
The local and convective acceleration terms in the full dynamic wave equation are important to the flood routing for flat or adverse slopes or very
steep slopes or unsteady flow conditions.
Only the full dynamic wave equation is applied in FLO-2D model and the contribution of each term in the equation is computed regardless of an
assumption of negligibility.
Solution Algorithm - How the Model Works
----------------------------------------
The differential form of the continuity and momentum equations in the FLO-2D model is solved with a central, finite difference numerical scheme.
This explicit algorithm solves the momentum equation for the flow velocity across the grid element boundary one element at a time.
The solution to the differential form of the continuity and momentum equations results from a discrete representation of the equation when applied at
a single point.
Explicit schemes are simple to formulate but usually are limited to small timesteps by strict numerical stability criteria.
Finite difference schemes can require lengthy computer runs to simulate steep rising or very slow rising floodwaves, channels with highly variable
cross sections, abrupt changes in slope, split flow and ponded flow areas.
The FLO-2D computational domain is discretized into uniform, square grid elements.
The computational procedure for overland flow involves calculating the discharge across each of the boundaries in the eight potential flow directions
(Figure 4) and begins with a linear estimate of the flow depth at the grid element boundary.
The estimated boundary flow depth is an average of the flow depths in the two grid elements that will be sharing discharge in one of the eight
directions.
Non-linear estimates of the boundary depth were attempted in previous versions of the model, but they did not significantly improve the results.
Other hydraulic parameters are also averaged between the two grid elements to compute the flow velocity including flow resistance (Manning’s n-value),
flow area, slope, water surface elevation and wetted perimeter.
The flow velocity (dependent variable) across the boundary is computed from the solution of the momentum equation (discussed below).
Using the average flow area between two elements, the discharge for each timestep is determined by multiplying the velocity times flow area.
The full dynamic wave equation is a second order, non-linear, partial differential equation.
To solve the equation for the flow velocity at a grid element boundary, initially the flow velocity is calculated with the diffusive wave equation
using the average water surface slope (bed slope plus pressure head gradient).
This velocity is then used as a first estimate (or a seed) in the second order Newton-Raphson tangent method to determine the roots of the full
dynamic wave equation (James, et.
al., 1977).
Manning’s equation is applied to compute the friction slope.
If the Newton-Raphson solution fails to converge after 3 iterations, the algorithm defaults to the diffusive wave solution.
In the full dynamic wave momentum equation, the local acceleration term is the difference in the velocity for the given flow direction over the
previous timestep.
The convective acceleration term is evaluated as the difference in the flow velocity across the grid element from the previous timestep.
For example, the local acceleration term (1/g*∂V/∂t) for grid element 251 in the east (2) direction converts to:
.. math::
:label:
\frac{\Delta (V_t - V_{t-1})_{251}}{g\, \Delta t}
where V\ :sub:`t` is the velocity in the east direction for grid element 251 at time t, V\ :sub:`t-1` is the velocity at the previous timestep (t-1)
in the east direction, ∆t is the timestep in seconds, and g is the acceleration due to gravity.
A similar construct for the convective acceleration term (V\ :sub:`x`/g*∂V/∂x) can be made where V\ :sub:`2` is the velocity in the east direction and V\ :sub:`4` is the velocity in the west direction for
grid element 251:
.. math::
:label:
\frac{V_2\, \Delta (V_2 - V_4)_{251}}{g\, \Delta x}
The discharge across the grid element boundary is computed by multiplying the velocity times the cross sectional flow area.
After the discharge is computed for all eight directions, the net change in discharge (sum of the discharge in the eight flow directions) in or out of
the grid element is multiplied by the timestep to determine the net change in the grid element water volume.
This net change in volume is then divided by the available surface area (A\ :sub:`surf` = storage area) on the grid element to obtain the increase or
decrease in flow depth ∆h for the timestep.
.. math::
:label: discharge_balance
\sum Q_x^{i+1} = Q_n + Q_e + Q_s + Q_w + Q_{ne} + Q_{se} + Q_{sw} + Q_{nw}
= A_{surf}\, \frac{\Delta h}{\Delta t}
where:
Q\ :sub:`x` = discharge across one boundary
A\ :sub:`surf` = surface area of one grid element
∆h/∆t = change in flow depth in a grid element during one timestep
The channel routing integration is performed in essentially the same manner except that the flow depth is a function of the channel cross section
geometry and there are usually only one upstream and one downstream channel grid element for sharing discharge.
The computational index is the flow direction (1 of 8 directions) not the grid element.
This simplifies and reduces the number of steps in the solution algorithm.
Each direction is visited only once during a sweep of the grid system domain and involves two grid elements whereas a grid element index requires each
grid element to be visited (Figure 4).
.. image:: img/Chapter2/Chapte002.jpg
*Figure 4.
Flow Direction.*
Flow Direction is the Computational Index not the Grid Element Number.
To summarize, the solution algorithm incorporates the following steps:
1. For a given flow direction location in the grid system, the average flow geometry, roughness and slope between two grid elements are computed.
2. The flow depth d\ :sub:`x` for computing the velocity across a grid boundary for the next timestep (i+1) is estimated from the previous timestep i
using a linear estimate (the average depth between two elements).
.. math::
:label:
d_x^{i+1} = d_x^{i} + d_{x+1}^{i}
3. The flow direction first velocity overland, 1-D channel or street estimate is computed using the diffusive wave equation.
The only unknown diffusive wave equation variable is the velocity.
4. The predicted diffusive wave velocity for the current timestep is used as a seed in the NewtonRaphson method to solve the full dynamic wave equation
for the velocity.
It should be noted that for hyperconcentrated sediment flows such as mud and debris flows, the velocity calculations include the additional viscous
and yield stress terms.
5. The discharge Q across the boundary is computed by multiplying the velocity by the cross sectional flow area.
For overland flow, the flow width is adjusted by the width reduction factors (WRFs).
The incremental discharge for the timestep across the eight boundaries (or upstream and downstream channel elements) are summed:
.. math::
:label:
\Delta Q_x^{i+1}
= Q_n + Q_e + Q_s + Q_w + Q_{ne} + Q_{se} + Q_{sw} + Q_{nw}
and the change in volume (net discharge x timestep) is distributed over the available storage area within the grid or channel element to determine an
incremental increase in the flow depth.
.. math::
:label:
\Delta d_x^{i+1}
= \frac{\Delta Q_x^{i+1}\, \Delta i}{A_{surf}}
where ΔQ\ :sub:`x` is the net change in discharge in the eight floodplain directions for the grid element for the timestep Δt between time i and i + 1.
See Figure 5.
6. The numerical stability criteria are then checked for the new flow depth.
If the Courant number stability criteria is exceeded, the timestep is reduced to the Courant number computed timestep, all the previous timestep
computations are discarded and the velocity computations begin again with the first computational flow direction.
7. The simulation progresses with increasing timesteps using a timestep algorithm until the stability criteria are exceeded again.
.. image:: img/Chapter2/Chapte003.jpg
*Figure 5.
Discharge Flux across Grid Element Boundaries.*
To accomplish the discharge exchange between grid elements based on the flow direction, the model sets up an array of side connections at runtime as
shown in Figure 6.
These flow directions are only accessed once during a timestep instead of the dual visitation required by searching for contiguous elements.
This approach facilitates additional parallel processing for model speedup and has the additional benefits of:
- Reducing the required discharge computations by about 40% increasing model speed.
- Ignoring completely blocked sides.
- Eliminating NOFLOC assignment for channels.
In this computation sequence, the grid system inflow discharge and rainfall are computed first, and then the channel flow is computed.
Next, if streets are being simulated, the street discharge is computed and finally, overland flow in 8-directions is determined (Figure 6).
After all the flow routing for these components has been completed, the numerical stability criteria are tested for every floodplain, channel or
street element.
If stability criteria of any element are exceeded, the timestep is reduced by various functions depending on the previous history of stability success
and the computation sequence is restarted.
If all the numerical stability criteria are successfully met, the timestep is increased for the next grid system computational sweep.
During a sweep of the grid system for a timestep, discharge flux is added to the inflow elements, flow velocity and discharge between grid elements
are computed and the change in storage volume in each grid element for both water and sediment are determined.
All the inflow volume, outflow volume, change in storage or loss from the grid system area are summed at the end of each time step and the volume
conservation is computed.
Results are written to the output files or to the screen at user specified output time intervals.
.. image:: img/Chapter2/Chapte009.jpg
*Figure 6.
FLO-2D Stability Criteria Flow Chart.*
The FLO-2D flood routing scheme proceeds on the basis that the timestep is sufficiently small to insure numerical stability (i.e.
no numerical surging).
The key to efficient finite difference flood routing is that numerical stability criteria limits the timestep to avoid numerical surging and yet
allows large enough timesteps to complete the simulation in a reasonable time.
FLO-2D has a variable timestep depending on whether the numerical stability criteria are not exceeded.
The numerical stability criteria are checked for every grid element on every timestep to ensure that the solution is stable.
If the numerical stability criteria are exceeded, the timestep is decreased and all the previous hydraulic computations for that timestep are
discarded.
Most explicit schemes are subject to the Courant-Friedrich-Lewy (CFL) condition for numerical stability (Jin and Fread, 1997).
The CFL condition relates the floodwave celerity to the model time and spatial increments.
The physical interpretation of the CFL condition is that a particle of fluid should not travel more than one spatial increment Δx (grid element side)
in one timestep Δt (Fletcher, 1990).
FLO-2D uses the CFL condition for the floodplain, channel and street routing.
The timestep Δt is limited by:
.. math::
:label:
\Delta t = \frac{C\, \Delta x}{(\beta V + c)}
where:
C = Courant number (C ≤ 1.0)
Δx = square grid element width or channel length
V = computed average cross section velocity
β = coefficient (5/3 for a wide channel)
c = computed wave celerity
While the coefficient C can vary from 0.2 to 1.0 depending on the type of explicit routing algorithm, a default value of 0.6 is recommended in the
FLO-2D model.
When C is set to 1.0, artificial or numerical diffusivity is theoretically zero for a linear convective equation (Fletcher, 1990).
If the simulation has some numerical surging identified by unreasonably high velocities or spikes in an outflow discharge hydrograph, the Courant
number should be reduced by 0.1 until a value of 0.2 or 0.3 is reached.
The Courant number is spatially variable by grid element within a small range.
With a successful completion the cell Courant number is permitted to increase by 0.0001 up to a value 0.05 greater than the assigned value.
If the Courant number is exceeded, the value is decrease by 0.002 until the assigned value is reached.
The assigned Courant is the minimum value allowed.
It may not be possible to completely avoid the artificial diffusivity or numerical dispersion using the Courant number to limit the timestep
(Fletcher, 1990).
Timesteps generally range from 0.1 second to 30 seconds with a typically timestep on the order of 1 second during the highest discharge flux.
The model starts with the a minimum timestep equal to 5 seconds and increases it until the numerical stability criteria exceeded, then the timestep is
reset to the computed Courant number timestep.
If the stability criteria continue to be exceeded, and the minimum timestep is not small enough to conserve volume or maintain numerical stability,
then the input data must be modified.
The timesteps are a function of the discharge flux for a given grid element and its size.
Small grid elements with a steep rising hydrograph and large peak discharge require small timesteps.
Accuracy is not compromised if small timesteps are used, but the runtime time can be long if the grid system is large.
The timestep incrementing and decrementing functions were designed to support the role of the Courant number in the model stability.
It was determined that varying the Courant number within a certain range of the original value reduced the number of ineffective timestep decreases.
In addition, the timestep increments and decrements were reduced to enable the computational timestep to more gradually adjust to numerical stability
criteria.
This replaced the method of a large timestep decrease followed by more numerous small timestep increases.
Results show there was a significant increase in runtime model speed up from 15 to 40%.
The stability criteria are applied as follows:
- Separate Courant number assignment for floodplain, channel and street flow.
- Automated spatial variation in the floodplain, channel or street Courant number within a specified range depending on the whether the Courant number
criteria was exceeded or not exceeded.
It should be noted that the obsolete stability parameters DEPTOL (% change in flow depth) and WAVEMAX (dynamic wave) stability criteria has been
eliminated.
A timestep accelerator parameter (TIME_ACCEL) was coded into the model.
Several adjustments of this variable have been made over the years.
The default value was changed from 0.1 to 1.0.
A higher value of TIME_ACCEL will result in larger timestep increments.
When the computational timestep is less than 1.0 second and a simulation timestep loop was successfully completed without exceeding the stability
criteria, the timestep is incremented by the TIME_ACCEL (default 1.0) + 0.001.
So if the timestep was 0.5, then the next timestep would be increased to 0.501 seconds.
If the timestep is greater than 1 second, then the timestep increment is:
.. math::
:label:
DSEC = DSEC + TIME\_ACCEL \times \frac{0.0085}{\text{XFAST}}
where:
DSEC = computational timestep in seconds
TIME_ACCEL = Coefficient to increase the rate of incremental timestep change.
Default value = 0.1. A value of 0.1 may result in a more stable simulation time.
A value of 0.2 or higher may result in a faster simulation.
XFAST = XFAST + 0.001 for each successfully completed timestep loop when DSEC > 1.0 second.
XFAST resets to 1.0 each time the DSEC timestep is decremented.
This algorithm increases the timestep uniformly until the timestep DSEC is greater than 1 second.
When DSEC > 1.0, successive increases in DSEC result in a larger value of XFAST which begins to slow down the timestep rate of change.
The maximum timestep is limited to 30 seconds.
The Importance of Volume Conservation
-------------------------------------
A review of any flood model simulation results begins with volume conservation.
Volume conservation indicates accuracy and numerical stability.
The inflow volume, outflow volume, changes in storage and losses (infiltration) are summed at the end of each time step.
FLO-2D volume conservation results are written to the output files or to the screen at user specified output time intervals.
Data errors, numerical instability, or poorly integrated components may cause a loss of volume conservation.
Any simulation not conserving volume should be revised.
It should be noted that volume conservation in any flood simulation is not exact.
While some numerical error is introduced by rounding numbers, approximations or interpolations (such as with rating tables), volume should be
conserved within a fraction of a percent of the inflow volume.
The user must decide on an acceptable level of error in the volume conservation.
While volume conservation with 0.001 percent is considered very accurate, most FLO-2D simulations have a volume conservation accuracy within a few
millionths of one percent.
Courant Number Variability for Num]erical Stability
---------------------------------------------------
The key to efficient FLO-2D flood routing is assigning numerical stability criteria that limits the timestep to avoid surging and yet allows large
enough timesteps for a fast simulation.
FLO-2D has a variable timestep that is a function of the numerical stability criteria.
The numerical stability criteria are checked for every grid element flow direction for each timestep.
If any of the numerical stability criteria are exceeded, the timestep is decreased and all the previous hydraulic computations for that timestep are
discarded.
The FLO-2D computation routing algorithm is an explicit scheme that is subject to the Courant-FriedrichLewy (CFL or Courant number) condition for
numerical stability.
The Courant number relates the floodwave movement to the model discretization in time and space.
The concept of the Courant number is that a particle of fluid should not travel more than one spatial increment Δx (between the center of cells) in
one timestep Δt.
If the model computational timestep exceeds the Courant relationship timestep, the stability criteria is exceeded and a timestep decrement occurs.
Mathematically the Courant relationship is given by:
.. math::
:label:
V + c = \frac{C\, \Delta x}{\Delta t}
where:
C = Courant Number (C ≤ 1.0)
Δx = FLO-2D square grid element width (distance between node centers)
V = depth averaged velocity
c = floodwave celerity = (gd)\ :sup:`0.5` where;
g = gravitation acceleration
d = the flow depth above the thalweg.
This equation relates the progression of the floodwave (V + c) to the discretized model in space and time.
The Courant number C can vary from 0.1 to 1.0, and a value of 1.0 will enable the model to have the largest possible timestep.
When C is set to 1.0, artificial or numerical diffusivity is theoretically zero for a linear convective equation.
Testing has shown that the FLO-2D model can run faster (more consistent higher timesteps) with greater stability if the Courant number is set to
values less than 1.0.
The FLO-2D default Courant number is 0.6 which will provide a numerically stable model for most applications.
Rearranging the CFL relationship, the model computes the Courant timestep Δt as:
.. math::
:label:
\Delta t = \frac{C\, \Delta x}{V + c}
Numerical instability occurs when the computational timestep is too large or the rate of change in the timestep is too large and too much volume
enters or leaves a grid element (discharge flux).
The corresponding change in flow depth can result in a high velocity (or Froude number) with the discharge causing a rapid fluctuation in the grid
element water surface elevation (surging).
Numerical surging may cause spikes in the discharge hydrograph (Figure 7), adverse water surface slope in the downstream direction or unreasonable
maximum velocities (VELTIMEFP.OUT (Figure 8) for the floodplain or VELTIMC.OUT for channels) or Froude numbers (SUPER.OUT).
Unreasonable maximum velocities are easy to identify because the VELTIM_x.OUT files are sorted in descending order.
The first several reported maximum velocities should not be significantly greater than the rest of the file as in the case of node 4887 in the
VELTIMFP.OUT file below.
.. image:: img/Chapter2/Chapte004.jpg
*Figure 7.
Numerical Surging in a Channel Element Hydrograph.*
.. image:: img/Chapter2/Chapte005.jpg
*Figure 8.
File Sample VELTIMFP.OUT.*
Using the Courant Number and Timestep Accelerator Parameter
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The global Courant Number is assigned in the TOLER.DAT file line 2 as follows:
.. raw:: html
Line 1. 0.1 00 (TOL; the DEPTOL parameter is not required)
Line 2. C 0.6 0.5 0.5 (floodplain, channel and street COURANT numbers)
Line 3. T 0.1 (TIME_ACCEL; referred to as ‘timestep accelerator’)
where C is a line character identifier and the Courant number for floodplain, channel and street are entered.
The Courant number should only be assigned for those components that are used.
The channel and street components may not be used in a given project.
The typical range of the Courant number is from 0.2 (slower more stable model) to 0.9 (faster less stable model).
The default of 0.6 is recommended as a starting value.
For models that appear to be unstable, reducing the Courant Number to a lower value in decrements of 0.1 (from 0.6 to 0.5 down to 0.2) should control
any numerical surging.
Lower Courant numbers should be assigned for steep rising hydrographs, high velocity conditions or high flow depths as in the case of a reservoir or
large detention basin filling.
Courant numbers are slightly spatially variable during runtime.
When the flood routing algorithm for a given grid element is completed successfully without a timestep decrement, the Courant number is increased by
0.0001 for the next timestep.
The increase in the Courant number is limited to 0.05 greater than the original global Courant number.
The Courant number cannot exceed 1.0 for any condition.
When the Courant number timestep is exceeded, then the Courant number is reduced by 0.002 and the model computational timestep is decremented by a
small percentage.
Consecutive timestep decrements force a larger percent reduction and after several successive decrements by the same grid element, the computational
timestep is set to the Courant number timestep.
In this manner, an initial small reduction in the computational timestep may allow the model to proceed with further exceeding the Courant timestep.
While the Courant number defines the base timestep, the TIME_ACCEL parameter controls the rate of increase of the timestep after a successful routing
loop is completed for a computational timestep.
The practical range of the TIME_ACCEL parameter is 0.1 to 1.0.
Values outside this range, although possible, would not improve the simulation.
If TIME_ACCEL = 0.1, a small rate of increase of the timestep occurs.
This reduces the number of total timestep decrements.
A large rate of change in the computational timestep for TIME_ACCEL = 1.0 will have a corresponding increase in the number of timestep decrements and
more opportunity for numerical instability.
The model speed tends to be the faster when there is balance between the number of timestep decreases and a gradual rate of climb or increase in the
computational timestep.
It is possible to limit the number of timestep decrements as reported to the TIME.OUT file, but this corresponds to a slow rate of increase in the
timestep.
A large rate of timestep increase (higher TIME_ACCEL) will result in higher average timesteps and more timestep decrements, but overall the model
speed may be faster and the runtime shorter.
**Suggested Approach:**
Use the default Courant number (0.6) and TIME_ACCEL (0.1) for the initial simulation.
Then review the VELTIMEFP.OUT file (and VELTIMEC.OUT file for the channel) files that report the maximum velocities in descending order to determine
if there are any unreasonable velocities.
Verify the adequacy of the results by reviewing the SUPER.OUT file for high maximum Froude numbers.
Post-processor program review of the graphical display of the channel hydrographs in the HYDROG or the PROFILES plot of the maximum channel water
surface elevation profile might also reveal numerical surging.
The following approach is suggested for making adjustments to speed up the model or slow down the model to eliminate numerical instability:
1. If the model has no numerical surging or unreasonable maximum velocities and it is desired to have the model run faster, increase the TIME_ACCEL
parameter from 0.1 to 0.25 and run the model again.
If the model runs faster and the results still don’t indicate any numerical instability, then increase TIME_ACCEL to 0.3 or 0.4.
Some numerical instability may begin to appear in VELTIMFP.OUT when TIME_ACCEL is 0.4 - 0.5 or higher.
2. Once unreasonable velocities or Froude numbers are noted, decrease the TIME_ACCEL by 0.05.
Review the mode runtime at the end of the SUMMARY.OUT file and do the various numerical instability checks.
At this point, a pattern will probably be apparent and the optimum Courant number and TIME_ACCEL parameter can be achieved.
If the model becomes numerically stable with the decreased TIME_ACCEL, it may be possible to increase the Courant number to achieve a slightly faster
model.
3. If the model has some initial numerical instability, leave the TIME_ACCEL at the default value of 0.1 and decrease the Courant Number from 0.6 to 0.5
to 0.4 over several runs until the numerical instability is eliminated.
Data adjustments to eliminate numerical surging might include increasing n-values using the limiting Froude number and the ROUGH.OUT file.
Depressed grid elements, hydraulic structure rating tables, deep ponded water or steep slopes may also contribute to unreasonably high maximum
velocities.
Following each flood simulation, the TIME.OUT file should be reviewed to determine which of the grid elements are frequently exceeding the Courant
timestep and contributing to a slow model speed.
For those grid elements with excessive timestep decrements, adjustments can be made to the node attributes such as topography, roughness, available
surface area (area reduction values ARFs) or width reduction values (WRFs).
Volume Evacuation from Small Grid Elements
------------------------------------------
Potential negative storage with very shallow flows was not a significant issue when the original TOL (depression storage TOL value) was typically
assigned a value of 0.1 ft.
Very low spatially variable TOL values are now being assigned for urban rainfall models to maximize runoff.
With the opportunity for faster FLO-2D simulations, several factors can contribute to the evacuation of volume from floodplain and channel elements
with shallow flow (negative storage volume).
Faster simulations and more parallel processing code have resulted in larger project grid systems with smaller grid elements.
Smaller elements combined with small assigned depression storage tolerance values (TOL ~ 0.004 ft or 0.001 m) results less volume on a cell for
shallow flows than for previous model versions (Figure 9).
The FLO-2D model has an explicit numerical scheme that exchanges flow with contiguous elements in eight directions, so it is possible with very
shallow flows to completely drain an element of volume if the outflow exceeds the inflow plus storage.
This may occur more frequently with hydraulic structure inflow nodes.
.. image:: img/Chapter2/Chapte006.jpg
*Figure 9.
TOL Definition.*
Grid Element Depression Storage that Must Be Filled Before Volume is Exchanged with Another Element
Previous Methods for Addressing Volume Evacuation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Any negative volume in floodplain or channel elements after the computational loop was previously redistributed to the other neighbor elements and the
volume in the given cell was set to zero.
This necessitated an extra computation loop for all the elements to re-sum all the volumes after the redistribution, but the timestep was not
decremented.
Instead, the normal computational routine continued uninterrupted but with an extra computational loop for all the grid elements for all timesteps.
This was a computationally expensive approach.
Addressing the volume evacuation issues required a review of the cells listed in the EVACUATEDFP.OUT and EVACUATEDCH.OUT files and adjusting cell
attributes such as n-values, elevation, surface area or other features or components.
Narrow inset low flow triangular channels were typically a problem and needed to be revised to a U-shaped channel that holds more volume at low flow
depths.
Correlating the grid elements listed in these files with those listed TIME.OUT enabled the most critical cells slowing down the model to be addressed
first.
In addition, if the number of times evacuated elements on the floodplain reached 1,000, this caused the model to stop; presuming that this number of
incidents of evacuated cells needed to be corrected.
Often evacuated cells may be associated with discharge or water surface controls such as hydraulic structure or other components.
The concept was to review the EVACUATEDxx.OUT files and adjust those evacuated element attributes before the final model product was completed.
Evacuated Element Revisions
~~~~~~~~~~~~~~~~~~~~~~~~~~~
Several codes revisions were made to simplify and automate the model response to evacuated elements including:
- The separate computational loop for evaluating the evacuated cells was eliminated.
- The limitation on the number of times (1,000) that evacuated elements were encountered was eliminated.
The number of times each floodplain or channel element is now evacuated is listed in the EVACUATEDxx.OUT files.
- Evacuated element n-values are increased by 0.001 to a limit of 0.250 for both the foodplain and channel.
- The model terminates the current timestep loop and reduces the timestep by 1 percent.
- The evacuated channel element TOL value is increased to 0.2 ft (0.067) if TOL is less than 0.2 ft (0.067 m) and increases the TOL by 0.02 ft (0.006 m)
if greater than 0.2 up to a maximum value of 0.4 ft (0.122 m).
- The floodplain evacuated element TOL value is increased to 0.1 ft (0.03) if TOL is less than 0.1 ft (0.03 m) and increases the TOL to 0.25 ft (0.076
m) if greater than 0.1 ft (0.03 m).
The focus is to adjust the flood hydraulics and the depression storage rather than the timestep for evacuated elements at shallow flows.
Reviewing the Evacuated Element Results
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
First review the SUMMARY.OUT and CHVOLUME.OUT file to see if there is any volume conservation error.
Volume conservation error is not always related to evacuated elements, but it could be.
Reviewing the EVACUATEDFP.OUT and EVACUATEDCH>OUT files and the TIME.OUT file will further help to identify if the model is not conserving volume in
response to the evacuated elements.
If EVACUATEDxx.OUT are empty or non-existent, the volume conservation error is being cause by some other component of the model.
If the grid elements in these files are correlated, then review the ROUGH.OUT file to see if the same elements have increased n-values.
When it appears that there are evacuated elements that are impacting the model performance, then it is necessary to review these elements in detail in
terms of their attributes (n-values, elevations) or components (ARF-values, streets, infiltration, etc.) and make some adjustments.
For these projects, only run the simulation until the model begins to slow down (a timestep less than 0.1 second or lower) and continue to adjust the
evacuated elements until these no longer appear in the output files.
Limiting Froude Number
----------------------
Using the Froude number in flood routing models is an important to the both the understanding of the floodwave movement and the numerical stability of
the model.
It is even more important when considering mobile bed channels such as alluvial fans or high bedload rivers.
The dimensionless Froude number Fr is given by:
.. math::
:label:
Fr = \alpha^{0.5}\, \frac{V}{(g d)^{0.5}}
Where:
V = depth averaged flow velocity
g = acceleration due to gravity
d = depth of flow above the thalweg
α = kinetic energy correction factor (involving the fluid density and specific weight).
Typically α is assumed to be 1.
The Froude number helps to define the influence of gravity on the flow pattern.
A low wave will propagate in free surface flow (open channel) depending only on the gravitational acceleration and the flow depth.
The movement (speed) of the shallow wave, known as the wave celerity c = √gd, is related to average flow velocity through the Froude number.
By accepting reasonable limitations of the overland (or channel) flow velocity and floodwave movement, the Froude number can be used to further define
the relationship between the velocity and flow depth.
For essentially steady and uniform flow, the Manning’s n value is defined would be defined by:
.. math::
:label:
n = \left( \frac{0.262}{Fr} \right) d^{0.17}\, S_o^{0.5}
indicating that the flow roughness is inversely proportional the Froude number.
By assuming a reasonable limiting Froude number, the n value can be estimated from the normal depth and slope for a given flow discharge.
In the FLO-2D model, the suggested n-value is based on either bankfull discharge for channel flow or 1 meter (3 ft) flow depth for overland flow
(roughness is fully submerged).
Suggested typical limiting Froude numbers are defined in Table 1.
*Table 1.
Range of Values Limiting Froude.*
.. list-table::
:widths: 30 30 40
:header-rows: 0
* - **Tool**
- **Flat or Mild Slope (large rivers and floodplains)**
- **Steep Slope (alluvial fans and watersheds)**
* - Channels
- 0.4 – 0.6
- 0.7 – 1.05
* - Overland flow
- 0.5 – 0.8
- 0.7 – 0.95
* - Streets
- 0.9 – 1.2
- 1.1 – 1.5
Similar values are also reported in the CVFED FLO-2D Application Guide.
If the limiting Froude number is exceeded, the grid element n-value increases by 0.001 for the next timestep.
When limiting Fr is no longer exceeded, the n-value decreases by 0.0005 if it’s greater than the original n-value.
The changes in n-value reported in ROUGH.OUT, FPLAIN.RGH, CHAN.RGH and STREET.RGH files.
The use of limiting Froude number in the FLO-2D model is documented in the FLO-2D Pocket Guide, the FLO-2D Reference Manual, and the CVFED FLO-2D
Application Guide.
The maximum n-values for discretized models will be greater than typical n-values for HEC-RAS cross sections (both channel and overbank areas).
This is because of the unsteady and non-uniform flow contribution between elements and the flow not being parallel to the cross section.
For flows over mobile bed conditions (supply unlimited), critical flows are approached asymptotically (Grant, 1997).
A relatively steep slope is required for flow with sediment transport to achieve critical flow because the flow hydraulics oscillate.
For Fr > 1, flow instability leads to rapid energy dissipation and bed erosion.
Flow is forced to stay around critical by incipient motion thresholds.
The equilibrium sand bed morphology tends to minimize the Froude number (Jia, 1990).
The limiting Froude number for mobile bed conditions can be approximated by (Grant, 1997):
.. math::
:label:
Fr = 3.85\, S^{0.33} \quad (\tau_{cr} = 0.03)
.. math::
:label:
Fr = 5.18\, S^{0.11} \quad (\tau_{cr} = 0.06)
Roughness n-values include many factors: n = n1 + n2 + n3 + n4 + …such as friction drag, vegetation, expansion/contraction, bed forms, flow in bends,
unsteady and non-uniform flow.
Shallow Flow Roughness and TOL
------------------------------
Open channel (or floodplain) uniform flow is characterized by a constant depth, velocity, flow area and discharge such that the bed slope, water
surface slope and energy grade line are all parallel.
Generally, uniform flow dictates that the flow is also steady.
Unsteady, uniform flow would not occur naturally.
For practical purposes natural uniform flow is also turbulent implying that a stable velocity distribution has been attained and the turbulent
boundary layer is fully developed.
There a number of uniform flow mean velocity equations for open channels and Manning’s equation is the best known of these:
.. math::
:label:
V = \frac{1.486}{n}\, R^{2/3} S^{1/2}
where
V = velocity,
R = hydraulic radius,
S = friction slope,
n = Manning’s roughness coefficient.
The hydraulic radius exponent value (0.667) has been known to vary over a range from about 0.59 to 0.85 depending primarily on channel geometry and
roughness (Chow, 1959).
The roughness coefficient or Manning’s n-value varies with a number of factors including but not limited to bed friction, bed form,
expansion/contraction, vegetation, obstructions, and flow depth.
Manning’s n-value is known to increase with decreasing flow depth.
Manning’s equation can be assumed to apply within and above the turbulent boundary layer.
By definition, the turbulent boundary terminates at depth where the vertical velocity distribution for uniform steady flow in a rough wide channel has
attained a value of 99% of the free stream flow.
At the lower depth, this flow regime extends from the height of the displacement thickness near the laminar boundary layer and varies as a function of
the Reynolds number.
The displacement thickness is generally 1/8 (rough) to 1/10 (smooth) of the turbulent boundary layer.
If the flow is very shallow roughness elements may protrude through the laminar sublayer and into the flow.
Flows are considered hydraulically rough if the grain size or roughness element is greater than 6 times the laminar boundary layer:
.. math::
:label:
\delta = 11.6\, \frac{\nu}{u_*}
where:
ν = kinematic viscosity
u\ :sub:`\*` = shear velocity
The applicability of Manning’s equation to a given flow condition depends on the relative submergence of the roughness elements (R/k\ :sub:`s`) where
k\ :sub:`s` is the effective roughness height.
In general, Manning’s equation is appropriate for a relative submergence (Julien, 1995):
.. math::
:label:
\frac{R}{k_s} > 100
which will correspond to the Manning-Stickler fixed bed roughness as function of sediment size D relationship (Simons and Senturk, 1976):
.. math::
:label:
n = \frac{D^{1/6}}{21.1}
For lower a submergence value (R/k\ :sub:`s` < 100), the logarithmic form of the resistance equation should be used (Julien, 1995).
For flow transporting sediment in suspension, the flow will be primarily turbulent if
.. math::
:label:
\frac{R}{k_s} > 70
Typical roughness height for grain size bed material can range from 0.0015 ft for rough concrete to 0.01 ft for coarse sand or uniform earth channels.
In this case Manning’s equation for a coarse sand plane bed would be applicable to about 0.7 ft.
Julien (1995) plotted the logarithmic velocity equation solution for turbulent flow over rough plane boundaries and it is noted that Manning’s
equation can still provide a reasonable approximation as low as about R/k\ :sub:`s` ~ 25.
Taking this as a less conservative approach for a rough surface where the roughness height is sand size or smaller, Manning’s equation should apply if
the flow depth is roughly 25 times the relative roughness.
For coarse sand this would be a flow depth of about 0.25 ft.
Manning’s equation will overpredict the velocity for shallow flow if typical low n-values assigned to deeper flows are applied.
Manning’s equation was an empirical equation that relates the velocity to several parameters using a coefficient (n-value) for flow depths generally
greater than 0.5 ft.
When computing velocity for shallow flow depths near the FLO-2D tolerance value TOL on the order of 0.1 ft or smaller, it is unlikely that the flow
will be fully developed turbulent flow.
In lieu of using multiple mean velocity equations, one for deeper flow and one for shallow flow, it is necessary to compensate for overpredicting the
velocity using Manning’s equation by assigning higher shallow n-values or using depth variable n-values or both.
Optimizing FLO-2D Simulations
-----------------------------
The FLO-2D model will run efficiently on essentially all new laptops and desktop computers that have a minimum of 8 processors and 16 GB of ram.
A typical minimum cost range for these computers would be ($800 - $1,000) for laptops and ($1,200 - $1,500) for desktops.
To enhance model speed and reduce model simulation runtime for large projects, generally bigger is better with more processors and RAM, but other
factors should be considered when selecting an off-the-shelf computer or have a computer custom built with selective features.
Your office may also be dedicating a portion of a server or group of linked computers in the office and we encourage you to work with your IT staff to
arrange the best allocation of computer resources.
Finally, there are a growing number of options for cloud computing, FLO-2D is currently exploring those with a cloud services provider to enhance
project containerization and fast, efficient model application and organization.
This document has some suggestions on computer specifications and how to determine if your system if operating efficiently.
Some guidelines on what output to review when comparing FLO-2D simulations are also presented.
There is brief discussion on what the future may bring in terms of FLO-2D enhancements.
We welcome your input and assistance in improving and expanding these recommendations.
Suggested Computer Specifications for FLO-2D Model Simulation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The processor speed is the most important computer component to reduce FLO-2D simulation runtimes.
When comparing computer runtimes with different numbers of more processor with varying clock speeds, processor speed may be the determining factor of
the overall computer model runtime.
Typically, FLO-2D model will run faster in computers with higher processor speed and slower in computers with lower processor speed regardless of the
number of processors over 24 to 32 cores.
When planning to purchase a new computer with a specific intention of running FLO-2D models, first review the processor speed (base clock and turbo
clock speeds) and then consider other computer features.
At this time, it is assumed that the processors and operating system will be 64-bit (32-bit computers can be used but will be decidedly slower and may
have conflicts with graphic DLL’s and preprocessor GUI’s).
Other considerations for computer hardware include a large RAM, multiple core processors and solid-state hard drives (SSD).
Solid state drives are important because booting up the computer system and launching FLO-2D models will take less time.
The SSD also increases the performance of a software program if the model has to frequently access (read/write) to the hard drive.
For very large FLO-2D projects, selecting an off-the-shelf computer is still appropriate and can save money over a custom-built computer.
Specifically, gaming computers have proven to be effective at decreasing the total simulation time because they are configured to maximize computing
and graphical display performance.
For FLO-2D urban simulations where the number of grid elements may exceed 2,000,000 with a lot of urban details (buildings, walls, etc.), a state-of-
the-art PC gaming computer in the price range of $2,000 to $3,500 is suggested.
One example of a 2019 off-the-shelf gaming computer has the following specs:
- Operating system: Windows 10 (64-bit)
- Processor: Intel Core i9-9900K Cores: 8 \| Threads: 16
- Process clock speed: 3.60 GHz (base clock); 5.0GHz (turbo clock) with overclocking
- Memory RAM: 32.0 GB or more Graphics: NVIDIA Graphic Card GeForce
The Flood Control District of Maricopa County recently tested two computers for comparison with the following specifications:
.. raw:: html
Computer 1:
CPU i7-8700k 6c @4.9GHz
16GB DDR4 RAM @2666MHz
1TB NVME PCIE SSD
Computer 2:
CPU i9-7940X 14c @3.9GHz
64GB DDR4 RAM @3000MHz
500GB NVME PCIE SSD
Multiple overclocks simulations were performed on both computers, pushing the overclock speeds limits while maintaining a safe overclock condition.
Computer 2 performed better when running the stock CPU and RAM settings.
Computer 1 performed better when overclocked and overall, first computer was able to finish larger models faster.
Simulations were performed by Riada/FLO-2D to compare some gaming computers.
The relative performance was assessed by comparing model runtimes for a large urban project simulation with 1,690,986 grid elements and all the
components active (storm drain, channels, buildings, rainfall, infiltration, walls, etc.).
The results were as follows:
I7 Intel processors - 4 cores, 8 processors with 3.5 GHz clock speed: 14.6 hrs runtime
I7 - 14 cores, 28 processors 3.3 GHz: 10.4 hrs runtime
I9 - 8 cores, 16 processors 3.60 GHz (base), 5.0 GHz (turbo w/overclocking): 7.8 hrs runtime
The I9 computer speedup was determined to be 1.4 and 1.9 times for these simulations.
Based on this testing, it is concluded that for the range of 32 to 64 processors, the reduction in simulation time that can be achieved by splitting
the workload between processors with parallel processing (OMP) code approaches the time that is consumed by splitting the instructions.
It is anticipated that above this number of processors, a reduction in the FLO-2D model simulation time will be best attained with higher processor
speed (GHz).
Amdahl’s Law determines the limit to improving the overall runtime speed for given parallelized portion of the code as the number of processors
increases (Figure 1).
The limiting factor is the cost of distributing the necessary data for sequential fractionalization of the instructions to higher orders of
processors.
.. image:: img/Chapter2/Chapte007.jpg
*Figure 10.
Model Speedup with Multiple Processors as a Function of the Parallelized Portion of the Code (en.wikipedia.org).*
There are other bottlenecks that may occur for a given system including certain models whose code does not conform to parallel processing, memory
bandwidth, and input/output bandwidth which may not scale with increased number of processors.
In these cases, adding more processors may lower the potential returns on speedup.
Most of the significant computational loops in FLO-2D have been parallelized and introducing further OMP code on smaller subroutine loops will achieve
only a few percent speed-up.
Our experience has been that more than 32 processor cores will not improve model speed on a conventional off-the-shelf desktop or laptop computer
system.
Cloud computing, advancements in processor clock speed and allocation of server resources are anticipated to provide the next level of FLO-2D model
runtime speed enhancement.
Assessing FLO-2D Model Computer Efficiency
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
“How do I know if my FLO-2D model simulation is optimally running on my computer? It seems to be slow.” This is a typical question/comment that is
posed by FLO-2D users.
There are several measures of model efficiency that can be reviewed.
In terms of the model output, the SUMMARY.OUT file will provide the simplest report of model speed at the end of the file.
The last two lines of SUMMARY.OUT report model runtime from the start of the flood routing computations to the end of the routing code along with date
and time of model termination:
COMPUTER RUN TIME IS: 14.25916 HRS
THIS OUTPUT FILE WAS TERMINATED ON: 2/28/2019 AT: 1:33:12
This computer runtime does not include the time to read the data files and set up the initialization of the variables before the model routing
algorithm begins.
It also doesn’t include the model computer time required to write the output to file and some output files are sorted in descending order which can
take 30 minutes or more for large complex projects.
When reviewing the SUMMARY.OUT file, the reporting of the volume conservation should be noted and if acceptable, then the runtime is representative of
the computation speed.
If there are volume conservation errors or numerical surging, the model is also using smaller computational timesteps and will report in a longer
runtime then when the model has satisfactory results.
Using the TIME.OUT, SUPER.OUT, VELTIMEFP.OUT and VELTIMEC.OUT files, it is possible to ascertain if there is numerical surging associated with a high
number of timestep decrements or unreasonable maximum velocities in the channel or on the floodplain.
A high runtime may be attributable to small timesteps and numerous timestep decrements and this would not be reflective of the potential model runtime
efficiency.
During a FLO-2D simulation, right-click the mouse on the task bar at the bottom of the computer screen and then click on the ‘Task Manager’ command to
view the model CPU usage.
Either the Processes Tab or the Performance Tab at the top of the Task Manager dialog box will report the model CPU usage.
If the CPU usage is 95% or greater, the model has access to all the processor cores and is running effectively.
If the CPU usage is less than 10% (typically 4 to 5%), the model is essentially running in serial mode and parallel processing is not being utilized.
You may be able to enhance this CPU usage during model simulation by clicking on the Details Tab, highlighting the FLOPRO.exe field, and right
clicking will display another dialog to set the runtime priority to ‘High’ as shown in Figure 2.
If you discover that the model is running at 10% CPU usage, however, there may be too many background programs running and rebooting the computer will
enable the FLO-2D simulation to have high priority and more computer resources.
A dedicated computer for FLO-2D simulations is recommended.
.. image:: img/Chapter2/Chapte008.jpg
*Figure 11.
Setting the FLO-2D Model Runtime Priority to ‘High’ in the Task Manager.*
Testing different builds of the FLO-2D model may yield slightly different results depending on the degree of OMP code in the model.
Ideally, the output differences using identical data between two FLO-2D builds is absolute zero.
This is result that we strive for.
The model versions provide identical results for serial code compilation, but the OMP code may cause non-significant differences between model
versions.
This is caused by how the different cores, threads or processors access the shared variables (data racing) that could lead to precision differences.
This is a programming error and can be alleviated by better code synchronization.
Technical support should be contacted in this case.
If the difference in maximum flow depth between two simulations using different model builds is only 0.05 ft (0.015 m) or smaller, this represents a
consistency level that is acceptable.
Even differences up to 0.5 ft (0.15 m) on a few grid elements is insignificant and this will be supported by the
SUMMARY.OUT file final volumes.
Larger differences and minor inconsistency in the results could be due to OMP data racing in specific data configurations and model components.
If unacceptable comparisons occur, reboot the computing and try to replicate the differences.
If so, document and report the differences to the FLO-2D staff and it will be immediately addressed.
In some cases with older model builds this may have already been resolved.
The Future of FLO-2D Model Optimization
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Within the framework of the desktop 64-bit operating system, only the processor clock speed will reduce the FLO-2D runtimes, however, the computers
sold today have top CPU speeds of about 3.6 GHz, which is essentially no different than a decade ago.
While computational intensive desktop computing is far from dead, Moore’s Law has essentially been put to rest.
Moore’s Law indicates that the number of transistors on a chip doubles about every 2 years, however, as processors became smaller and more numerous,
the drive for better performance became a fight against heat transfer.
The development of multiple core chips also required implementing OMP code and encountered the limitations associated with Amdahl’s Law as previously
discussed.
The utilization of workstation servers with greater performance flexibility will continue to expand, but the server trend for the next ten years will
be to achieve better performance using less power while trying to run more threads concurrently.
The ultimate future of flood modeling will be in cloud computing.
There are also some new technologies that may impact personal computing: Quantum Computing; and Graphene-based CPUs which will be mentioned below.
Cloud computing will negate the need for the cycle of upgrading personal and corporate computers and servers.
It will use the technology of containerization; the processing and storage of data in the same location as the computing device.
In essence, the cloud processing power will replace both the computer's hard drive and processor, and all that will be needed is a small basic laptop
or desktop computer.
There is also the financial incentive of decreasing incremental costs to perform large model simulations.
The adoption of cloud computing by corporations and agencies will be the result of accessibility by multiple users, ease of use and scalability.
This will also lead to the eventual progression to real-time modeling and streaming for event driven early flood warnings.
For agencies administer flood projects, cloud computing will simplify the process of monitoring data resource allocation and contractor services
delivery with a much higher level of confidence.
Cloud security and compliance will remain the shared responsibility of all the stakeholders, not the cloud service provider.
This is important for the application of the FLO-2D model within a cloud system where the integrity of simulations must be ensured.
FLO-2D is working with a cloud service company to provide the model users with a simple and scalable solution for running their simulations in the
cloud.
This framework has the benefits of access to powerful multi-core machines, parallel processing, batch processing, easy collaboration, and results that
are easy to share with staff, clients and agencies.
Testing this process has already commenced for projects of various sizes and complexity.
FLO-2D is also researching compiling the model executable for Linux operating systems to further reduce the cost of cloud computing.
Other technologies that FLO-2D is following that could transform intensive model computing is quantum computing and graphene computing.
Quantum computers use the principles of quantum physics of subatomic particles to perform algorithm calculations and process massive datasets.
Data is stored in quantum bit (qubits), a simpler form of data that speeds up processing by storing conventional data bits and bytes at the same time.
Qubit-based computers process data thousands of times faster compared to conventional computers, enabling Monte Carlo simulations and forecasting in
real time.
Quantum computers will also be very small, on the order of the atom.
At the present time, they are expensive, and their use requires special equipment.
Another technology is graphene computers which will have chips with a super-conducting material capable of heat dissipation faster than any other
chip, while at the same time conducting electricity two hundred times faster than silicon.
Graphene based CPUs would achieve a speed a thousand times higher clock speed with a hundred times less energy consumption.
At the present time, however, graphene computers are still on the drawing board.
FLO-2D is keeping abreast of these developments to achieve the best modeling speed and efficiency while being the lowest cost commercial flood model
available.