Fluid Dynamics Formulas

Formulas used in the Virtual World. For more theory 
General assumptions:

  1. Flow is turbulent.
    So formulas of Manning, Chezy and Darcy-Weisbach can be used.
  2. Flow is sub-critical.
    So calculations of water-levels start downstream and goes upstream.
  3. Flow is steady
    Discharge is not changing rapidly. Of-course over time discharge (and thus water-levels etc) can change.
  4. Sea is salt water, on the island fresh water
  5. There are two main watersystems. The watersystem on the island and the sea (waves, tide, current). The connection between the two are by the culvert in the dike and seepage. High tide in combination with low water level on the island could make water flow from sea to the island. High tide can cause seepage to the land, low tide seepage to sea. Maybe in high tide in combination with big waves can cause water flowing over the dike to the island.
  6. Water enters the systeem by: rainfall and seepage.
  7. Water leaves the island through : Culvert in dike, pumping station, seepage and evapotranspiration.
  8. System is dendritic.
    So water has one direction to flow.

Calculations in FD_watersystems

  • For each branch waterlevel (ylevel) and Energylevel (Hlevel) are calculated at the beginning and end of the branch.
  • The Hlevel downstreams is transferred from the Hlevel upstreams branch downstreams. The ylevels are calculated with the dimensions from the branch. So it could be that ylevels between 2 branches are not the same? When drawing the yline, a transition between the 2 y levels has to be made.
  • For connector you have different width and slope downstream and upstream!!
  • CulvertRectangle: If downstreams  waterdepth < Height culvert : Type is converted to Trapezium. Note: With high velocities, water level will rise and the type maybe should be concerted back to culvertrectangle. This can be solved by using several extra points.
  • Water level at weir. 
  • In a branch (between 2 points) discharge is always the same. 
  • Normally HlevelWorld_m. x = HlevelWorld_m.y downstream branch. Except with a weir.
  • At a point with InputDischarge, the waterlevel will drop because the increase of velocityhead. If this happens very suddenly is looks odd. Solution is to add 1 or more extra point downstream of a point with InputDischarge. So the full velocityhead is taken into account at the point downstream of the pint where InputDischarge occurs.
  • 2 special branches when calculating. First is the outflow: Discharge = 0. Value in InputDischarge.x is the velocity in the receiving water. Second is the weir : There is a jump in HlevelWorld compared to the downstream waterlevel.
  • Weir is always assumed as free flow.  And as a short crest. Waterlevel .x and .y are the same.

Workflow calculation

In the case of type = trapezium

  1.  HlevelWorld_m.x = HlevelWorld.y previous branch
  2. yLevelWorld_m.x = yLevelWorld.y previous branch 
  3. Get Discharge from point upstreams
  4. Calculate Depth_m.x ,WettedArea.x, WetterPerimeter.x, HydraulicRadius.x, Velocity.x
  5. Calculate Hslope, dH, Time
  6. HlevelWorld_m.y = HlevelWord_m.x + dH
    ylevelWorld_m.y = yLevelWorld,x + dH
  7. Calculate Depth_m.y ,WettedArea.y, Velocity.y, VelocityHead
    ylevelWorld_m.y= HLevelWorld_m.y – VelocityHead
  1. Outflow down-streams is calculated separate
  2. Check if branch is weir. If weir calculate H above weir, convert to HlevelWorld_m.x = HlevelWorld.y.  yLevelWorld_m = CrestlevelLevelWorld_m + 2/3 * H.
  3. If branch is not weir
  4. If branch = connector; calculate average width and average slope
  5. Calculate Depth_m.x ,WettedArea.x, WetterPerimeter.x, HydraulicRadius.x, Velocity.x
  6. Calculate Hslope, dH, Time
  7. HlevelWorld_m.y = HlevelWord_m.x + dH
  8. ylevelWorld_m.y = yLevelWorld,x + dH
  9. Calculate Depth_m.y ,WettedArea.y, Velocity.y, VelocityHead
  10. ylevelWorld_m.y= HLevelWorld_m.y – VelocityHead

Formulas

Discharge
Bernoulli
Area pipe
Carnot

Chezy coefficient
Chezy energy slope

Manning energy slope
Culvert
Slope energy line
Sharp crest free flow
Partially filled pipes
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