Let us examine how to introduce singularities in the double sweep process.

Figure 27

The problem to be solved in the case of a singularity is the following:

[38]

We need to transmit the impedance relation:

Rj.DQj + Sj.DZj = Tj

to the downstream cross section of the singularity.

We shall assume that the device is moveable, and that variation law W(t) is known a-priori.

The device equation can be written at the instant t + (n + 1)dt:

Qin+1 = f(Zin+1, Zjn+1, Wn+1)

Then:
DQi = f(Zin + DZi, Zjn + DZj, Wn+1) - Qin

An expression of the non linear impedance relation is obtained in the following form:

DQj = f(Zin + ( - DQj), Zjn+DZj, Wn+1) - Qin [39]

The problem is then to find the best possible linear approximation to this expression. The tangential approximation equation of the device variation law can be written as:

Qin + DQi = f(Zin + DZi, Zjn + DZj, Wn+1)

= f(Zin, Zjn, Wn+1) + (Zin, Zjn, Wn+1) DZi

+ (Zin, Zjn, Wn+1) DZj

which leads to: Rj.DQj + Sj.DZj = Tj [40]

with:

This method cannot avoid the tangential approximation error of the device variation law created at each time step n, but counterbalances it in the next time step n+1. Indeed, a "correction wave" is included in the expression of the Tj coefficient in the form of an additive term Si ( f() - Qin ).

Nevertheless, errors due to the tangential approximation can be significant in the case of rapid variations of flow conditions (device operations, free-flow to submerged flow transition, etc.). It is then necessary to have as precise an estimation as possible of the evolution of the two variable Zi and Zj during time step Dt. So, variables Z_i, Z_j, Q_i, and Q_j are calculated locally, and Z* and Q* estimated by linear approximation by the following procedure. At each singular section, three equations [38] are available, with four unknowns.

So, a fourth equation is needed in order to calculate Z_i, Z_j, Q_i, and Q_j locally. This equation is an assumption on Zj. This assumption does not really attempt to get close to the missing R’S’T’ equation but rather to its effects on the evolution of the Zj value.

Let’s assume that:

DZj = k.DQj [41]

k being determined during the previous time step.

The following procedure for the computation of Rj Sj Tj is used:

1) Hypothesis on Zj ([41]) + [40] = expected DZj* value

=> expected DQj* and DZj* values

2) Computation of two expected values DQi* and DZi*

=> DQi* = DQj*
and expected DZi* value thanks to the RiSiTi impedance relation.

We assume that the real values DQi, DZi and DZj will be close to DQi*, DZi* and DZj*.

This results in:

We can then write:

Qin+1 = Qin + DQi = f(Zin + DZi, Zjn + DZj, Wn+1)

= f(Zin + DZi* + dzi, Zjn + DZj* + dzj, Wn+1)

with small values for dzi and dzj.

If we set: f(*) = f(Zin + DZi*, Zjn + DZj*, Wn+1)

we obtain:

Qin + DQi = f(*) + dzi + dzj

=> DQi = f(*) - Qin + (DZi - DZi*) + (DZj - DZj*)
= f(*) - Qin - ( DZi* + DZj*) + DZi + DZj

By adding:

we get:

[42]

with:
D = f(*) - Qin - ( DZi* + DZj*)

and f(*) = f(Zin + DZi*, Zjn + DZj*, Wn+1)

Let us take a closer look at how equation [41] is determined: let’s assume that Zj varies between n and n+1 in the same way as between n-1 and n, with respect to variation of flow.

Then:

k =

Set k = 0 if | Qjn - Qjn-1 | < 0.01 or if the slope of the downstream R’S’T’ has the same sign as the upstream RST equation.

Let us examine how to verify whether the RST and the R’S’T’ equations have slopes of opposite signs:

The system composed of equations [38] and [41] is written as follows:

[43]

DQj* = DZi* + DZj*

=> DQj* = ( - .DQj*) + DZj*

=> DQj* (1 + ) = + DZj*

the system [43] is:

It is necessary that: k () < 0

or k (1 + ) < 0

in order that the RST and the R’S’T’ have slopes of opposite signs.