Equations in Mohid Land - Revision history

Davidbrito: /* 2D descritization (Runoff) */

2013-01-11T10:34:38Z

‎2D descritization (Runoff)

Davidbrito: /* Surface Water (Runoff and Drainage Network) */

2013-01-11T10:26:18Z

‎Surface Water (Runoff and Drainage Network)

Davidbrito: /* Surface Water (Runoff and Drainage Network) */

2013-01-11T10:23:46Z

‎Surface Water (Runoff and Drainage Network)

Davidbrito: /* Moment Equation or "How those Properties Move" */

2013-01-11T10:10:43Z

‎Moment Equation or "How those Properties Move"

Davidbrito: /* Mass Conservation Equation or "How Properties can't have black holes" */

2013-01-11T09:59:11Z

‎Mass Conservation Equation or "How Properties can't have black holes"

Davidbrito: /* Surface Water (Runoff and Drainage Network) */

2012-10-12T09:03:52Z

‎Surface Water (Runoff and Drainage Network)

Davidbrito: /* Simple Explanation of St. Venant Terms */

2012-10-12T08:51:54Z

‎Simple Explanation of St. Venant Terms

Davidbrito: /* 2D descritization (Runoff) */

2012-10-11T15:55:58Z

‎2D descritization (Runoff)

Davidbrito: /* 1D descritization (River) */

2012-10-11T15:54:50Z

‎1D descritization (River)

Davidbrito: /* 2D descritization (Runoff) */

2012-10-11T15:29:46Z

‎2D descritization (Runoff)

@@ Line 151: / Line 151: @@
 ====2D descritization (Runoff)====
 St. Venant equation in 2D needs to consider that u and v "cells" have fluxes in x and y direction (see [[Grids]]):
-[[Image:EqnQntMov StVenant 2D x.png|1000px|thumb|center|]]
+[[Image:EqnQntMov StVenant 2D x2.png|1000px|thumb|center|]]
-[[Image:EqnQntMov StVenant 2D y.png|1000px|thumb|center|]]
+[[Image:EqnQntMov StVenant 2D y2.png|1000px|thumb|center|]]
 and descritizing to cells and time step:
-[[Image:EqnQntMov StVenant 2D x disc.png|1000px|thumb|center|]]
+[[Image:EqnQntMov StVenant 2D x disc2.png|1000px|thumb|center|]]
-[[Image:EqnQntMov StVenant 2D y disc.png|1000px|thumb|center|]]
+[[Image:EqnQntMov StVenant 2D y disc2.png|1000px|thumb|center|]]
   where i and i+1 and j and j+1 are u and v cell faces (see [[Grids]])

@@ Line 127: / Line 127: @@
 And the same equation with flow (multiplied by vertical area):
-[[Image:EqnQntMov StVenantQ 2.png|1000px|thumb|center|]]
+[[Image:EqnQntMov StVenantQ 3.png|1000px|thumb|center|]]
 ====1D descritization (River)====

@@ Line 115: / Line 115: @@
 And the complete set for momentum equation (St. Venant Equation in 1D or 2D):
-[[Image:EqnQntMov StVenant1.png|1000px|thumb|center|]]
+[[Image:EqnQntMov StVenant2.png|1000px|thumb|center|]]
   where:

@@ Line 78: / Line 78: @@
 The Newton's 2nd law and using density:
-[[Image:EqnQntMov Newton Masses.png|1000px|thumb|center|]]
+[[Image:EqnQntMov Newton Masses 2.png|1000px|thumb|center|]]
   where: x_i = spatial direction where velocity is solved
   x_j = all the spatial directions (that interact with transmission of momentum)

@@ Line 25: / Line 25: @@
 The property transport equation in "infinitesimal volume" form::
-[[Image:EqnPropertyTransport_InifinitesimalVolume.png|1000px|thumb|center|]]
+[[Image:EqnPropertyTransport_InifinitesimalVolume1.png|1000px|thumb|center|]]
 where:

@@ Line 182: / Line 182: @@
 Here it can be seen that the influence of the two acceleration terms (inertia and advection) in the computation:
-) inertia will add to the equations that the new velocity in each cell will depend on the old velocity in that cell and of course on the effect of the forces. This means that when acceleration exists, inertia in the model will take into account that even if the forces may be trying to push the water in the opposite direction that it was moving, the last velocity is important. And bigger the last velocity, the lower weight is given to the forces and more weight is given to the last velocity. Basically, bigger the last velocity bigger the tendency for the movement to continue with a similar velocity and minimizing the effect of the resulting forces (inertia).
+) inertia will add to the equations that the new velocity in each cell will depend on the old velocity in that cell and of course on the effect of the forces. This means that when acceleration exists, inertia in the model will take into account that even if the forces may be trying to push the water in the opposite direction that it was moving, the last velocity is important. And bigger the last velocity, the lower weight is given to the forces and more weight is given to the last velocity. Basically, bigger the last velocity bigger the tendency for the movement to continue with a similar velocity and minimizing the effect of the resulting forces (inertia). This term is the "But I was going that way" term.
-) advection will add to the equation that the new velocity will depend on the momentum arriving or leaving to the neighbours. This means that if the water is flowing fast downhill, momentum will continue to be transported even if the slope changes to flat areas and the level gradient gets small (would result in low flows downstream and higher water columns if acceleration terms are not computed). Basically with big slope changes, the water will tend to continue the movement in the flow direction mimimizing the effect of the resulting forces (tath may even be pointing in the opposite direction).
+) advection will add to the equation that the new velocity will depend on the momentum arriving or leaving to the neighbours. This means that if the water is flowing fast downhill, momentum will continue to be transported even if the slope changes to flat areas and the level gradient gets small (would result in low flows downstream and higher water columns if acceleration terms are not computed). Basically with big slope changes, the water will tend to continue the movement in the flow direction mimimizing the effect of the resulting forces (tath may even be pointing in the opposite direction). This term is the "But they are pushing/stoping me in that way" term.
-If inertia and advection are not used (or no acceleration) then forces balance themselves and the velocity is solely dependent on the forces and the last velocity or the transport of momentum from the neighbours is not accounted. Meaning that if forces push water from one time step to other in opossite directions water will instantaneously change directions and velocity. Moreover in case of changing slopes abruptly (from high to low), the lack of acceleration terms and because flow lowers downstream (lower hydraulic gradient in flat areas) in order to balance the reduction of flow the water column rises in unrealist way (the lack of momentum advection puts the energy dissipation in friction).
+If inertia and advection are not used (or no acceleration) then forces balance themselves and the velocity is solely dependent on the forces and the last velocity or the transport of momentum from the neighbours is not accounted. Meaning that if forces push water from one time step to other in opossite directions water will instantaneously change directions and velocity. Moreover in case of changing slopes abruptly (from high to low), the lack of acceleration terms and because flow lowers downstream (lower surface gradient in flat areas) in order to balance the unreal reduction of flow the water column rises in unrealist way (the lack of momentum advection puts the energy dissipation in friction).
 An important application for using inertia and advection is when slope changes abruptly from hilly to flat.

@@ Line 137: / Line 137: @@
   u subscript is from upstream
   d subscript is from downstream
 And putting it relative to new flow <math>Q^{t+\Delta t}</math>: