PERMANENT FLOW HYDRAULICAL VERIFICATION

# PERMANENT MOTION STATE HYDRAULICAL VERIFICATION

The permanent motion with gradual variations flowing is that which is occurs in a free current with  gradual variations in the beds cross section. continuity determines that capacity is constant, while  velocity and cross section vary gradually along the flow’s s axis.

If we consider a permanent flowing current with a minor inclination and a constant Q flow, isolating a riverbed section with ads length (with the s abscissa measured from a point that was not previously calculated, in the horizontal direction towards the coincident with the flowing point) the depth is being subsided with if ds and the total load line withJds , indicating the fall with J.

From simple geometrical considerations we have:

The expression b), in which for simplicity if was specified with as being equal to i, it indicates the total specific energy that refers to the depth, grows by the depth’s subside and is diminutions through the resistance effect.

Substituting the finite growth differences b) becomes:

E results depending on s, through the h height we have:

From E' s displacement depending on h, we know that E is decreasing and increasing in function of h (dE/dh < 0) for rapid currents (h < k), and is increasing (dE/dh > 0) for lent currents (h > k); in function of the critical state dE/dh = 0.

Accepting for the charge loss J the following formula:

results that J is as decreased as h is increased, assuming that all the dinominater's terms are increasing with h.

c)'s numerator is positive (i > J) for water heights that surpass those of the uniform flow (h > h0), negative for water  heights that no not surpass those of the uniform flow(h < h0).

The considerations made until now are at the base of the water profile's permanent flow represents: we note that when the water height is approaches that of the uniform flow, dh/ds tends to zero, or the profile tends to lie parallel to the bed and to the uniform flow.

But when the height is tending to the critical value k, dE/dh tends to be annulated so the profile tends to be displayed  perpendicularly on the base (depth).

So the permanent flow profiles are analytically described by the expression b), so specify at least one arbitrary constant to determine by defining a section's height like this h = h*.

Such a condition is calculated in function of a perturbatory cause that provokes, in a section, a height h different from that of the uniform flow; for this purpose the perturbatory cause can have influence uphill only if the current is slow (or becomes slow), it can have influence downhill only if the current is a fast one (or it becomes rapid because of this influence).

All these get to the conclusion that the profile's relation - the displacement's depart point - is found downhill if the current is slow and uphill if the current is fast.

In such a section we have to note the height h* determined as the cause of perturbation and the level difference h*- h0 will be known in reference to uniform flow.

How the program computes the profile

The program computes the profile with the standard step method: between two consecutive sections it imposes the energy balance (water surface elevation plus velocity head), with friction losses evaluated with the average friction slope between the two sections (Chézy formula with the Gauckler-Strickler coefficient, main channel and overbanks treated separately). The water depth in the next section is obtained by solving the equation directly: there is no scanning step to assign. When two sections are far apart, the reach is divided into sub-steps.

The calculation is in mixed flow regime:

1. starting from the downstream condition, the subcritical profile is computed moving upstream;
2. starting from the upstream condition, the supercritical profile is computed moving downstream;
3. in each section the total force (hydrostatic force plus momentum) of the two profiles is compared and the profile with the larger force is kept. Where supercritical flow prevails the flow stays supercritical; where subcritical flow prevails a hydraulic jump forms.

The calculation chooses between subcritical and supercritical flow, section by section. The two boundary conditions are still needed: without the downstream condition there is no starting point for subcritical flow, without the upstream one there is no starting point for supercritical flow.

If in a reach the energy equation has no solution, critical depth is assumed in the section and the program reports it among the warnings at the end of the calculation.

Boundary conditions

Upstream or downstream? There is no option to choose whether to start from upstream or downstream, and none is needed. The program computes supercritical flow starting from upstream and subcritical flow starting from downstream, then keeps the right one in each section: in the same river some reaches turn out supercritical (usually the steep ones) and others subcritical, and where the flow changes from supercritical to subcritical a hydraulic jump forms. This is why both conditions are always assigned: the upstream one is the starting point of supercritical flow, the downstream one of subcritical flow. For each end you only state what the water is like. Imposing a single direction of calculation in advance would mean already knowing that the flow is subcritical everywhere or supercritical everywhere; if it is not, the profile is wrong along the whole reach.

The conditions are assigned in Hydraulic checks > Boundary conditions (or Computation options > Computation options), separately for downstream and upstream and for each return period.

Choice	Downstream (subcritical flow)	Upstream (supercritical flow)
Uniform flow	The right choice in most cases, when nothing downstream controls the water level. The program uses the uniform flow depth with the bed slope.	The usual choice.
Imposed depth	Only if the downstream level is known: sea, lake, confluence with a larger river, weir.	Only if measured and lower than critical depth.
Critical depth	If the downstream section is at a drop, a sill or a free outfall.	If upstream there is the outlet of a lake or reservoir, or a bed drop.

Good to know:

• downstream, an imposed depth lower than critical is not compatible with subcritical flow, and upstream an imposed depth higher than critical is not compatible with supercritical flow: in both cases the calculation uses critical depth. The program warns about it when the window is confirmed;
• with imposed depth and all depths equal to 0 the calculation uses critical depth; in this case the program opens the window before the calculation so that you can check them;
• in mild reaches, where the flow is subcritical, the upstream condition does not change the result: it matters in steep reaches;
• the effect of an uncertain downstream condition fades moving upstream: place the downstream section a little beyond the reach of interest;
• if you are not sure what to choose, use uniform flow both downstream and upstream: it is the setting of new projects.

After the calculation the status bar shows the conditions used (for example "downstream uniform flow, upstream critical depth").