Kalinin-Miljukov method, basic principles

The Kalinin-Miljukov-method (KM-method) is based on the principle of the reservoir cascade. The river stretch is subdivided into characteristic sections that are considered as individual reservoirs (see Figure 7.16). The problem of instationary open channel flow is therefore reduced to the successive computation of reservoir retention. A special form of this model is a cascade of identical, linear reservoirs. (Nash-cascade, see Chapter 7.4.4 and 5.3.3).

river stretch

outflow inflow

reservoir cascade model

time t time t

outflow Qz outflow Qz

Figure 7.16: Kalinin-Miljukov method, instationary open channel flow described by a reservoir cascade model

Kalinin and Miljukov established the theoretical basis and developed a method to derive the necessary parameters from coarse geometric and hydraulic channel data without in- or outflow measurements. The KM-method requires a certain homogeneity of the river stretch or at least of distinct parts of it. In addition, the prerequisites of the hydrodynamic equations in Chapter 7.3 must be fulfilled.

• hydraulically similar river cross-sections (similar shape and roughness throughout the whole section),

• uniform bed slope (no drops or water breaks).

The method derives from Manning’s equation of discharge:

2 1 so-called water stage -outflow relation Q_stat(h).

( ) ( ) ( )

^{23 12}

stat St hy stat

Q h =A h k⋅ ⋅r h ⋅I (7.44)

Q_stat [m³/s] stationary open channel outflow,

h [m] water depth,

k_St [m^1/3/s] Strickler’s roughness coefficient, A [m²] cross-sectional area,

r_hy [m] hydraulic radius, I_E [-] energy line gradient,

I_stat [-] energy line gradient for stationary flow, I_so [-] river bed slope.

Manning’s equation provides only a rough estimation of the outflow hydrograph or the water depth -outflow relation. Preferably the outflow should be measured directly. For instationary open channel flow it can be assumed that for identical water depths the flow parameters remain the same. Therefore the only difference between stationary outflow Q_stat and instationary outflow Q_inst is the difference in the energy lines. In contrast to the uniform gradient of the stationary energy line, the gradient of the instationary energy line may change in the course of a flood.

( ) ( ) ( )

^{23 12}

inst St hy inst

Q h = A h k⋅ ⋅r h ⋅I (7.45)

Q [m³/s] instationary open channel flow,

I_inst [1] energy line gradient for instationary open channel flow.

Under the premise that even in the case of instationary flow the energy line is parallel to the water surface line, the gradient of the instationary energy line I_inst consists of a stationary part I_stat and an additional part ∆I (see Figure 7.17).

inst Sp

inst stat

I I

I I I

=

= + ∆ even for instationary flow (7.46)

I_Sp [1] slope of water surface,

∆I [1] gradient difference between stationary and instationary energy line gradient.

Figure 7.17: Longitudinal river cross-section, comparison of stationary and instationary energy line gradient (I_stat and I_inst) for flood rise

Inserting this in the discharge equation provides:

( ) ( ) ( )

²³

inst St hy stat

Q h = A h k⋅ ⋅r h ⋅ I + ∆ I (7.47)

The root on the equation can be rearranged and transformed to a power function. It can be proved that already the first two elements of the function provide sufficient precision because the slope angles are usually very low (the error is usually lower than 6 %).

1 certain differential value ∆Q.

( ) ( ) ( )

stationary case with a water depth h + ∆h that differs in height by ∆h.

( ) ( )

inst stat

Q h =Q h+ ∆h (7.51)

∆h [m] difference between water depths for identical outflow in stationary and instationary case

outflow Q [m³/s]

flood rise flood

recession

water depth h [m] stationary outflow curve

Figure 7.18: Relation between water depth and outflow for stationary and instationary flow (Q_stat(h) and Q_inst(h))

Considering this, it becomes obvious that the water stage-outflow curve Q_stat(h) may be applied to determine instationary outflow Q_inst , if the water depth h is replaced by the sum of h + ∆h (see Figure 7.18).

The outflow difference ∆Q can be determined from the water stage-outflow curve as the difference between the stationary outflows at the water depths h and h + ∆h (see Figure 7.18).

( )

^inst

( )

^stat

( )

^stat

( )

^stat

( )

^stat

( )

Q h Q h Q h Q h h Q h Q h

∆ = − = + ∆ − = ∆ (7.52)

∆qstat [m³/s] outflow difference related to the stationary water stage - outflow curve Another simplification the KM-method takes advantage of is the assumption of a linear water surface. If this assumption is also considered valid for instationary flow, then geometric relations between the water depths at the individual cross-sections can be determined (see Figure 7.19). Starting at a cross-section r with a water depth h, the same water depth can be found upstream in the cross-section m at the distance l_b.

Figure 7.19: Longitudinal cross-section through a river stretch with the characteristic length l_ch, water surface and reference cross-sections r and m

b

b

h I l or I h

l

∆ = ∆ ⋅ ∆ = ∆ (7.53)

l_b [m] reference stretch in the channel

If l_b is known, the instationary outflow in the cross-section can be correlated with a stationary water depth at the cross-section. In other words, the stationary water stage - outflow curve indicates even instationary outflow. Therefore water depth and outflow are related directly, with the only problem being that the relation occurs between two different cross-sections. The stationary water depth that is associated with an instationary outflow occurs simultaneously at a certain distance l_b further upstream. The stationary water stage - outflow curve may for this reason be applied in the case of instationary flow also. Hence the difficulties with the instationary outflow loop can be ignored. This relation is only valid if the cross-sectional shape and the outflow curve do not change along the river stretch.

( ) ( )

inst stat

Q h =Q h+ ∆h (7.54)

cross-section r cross-section m

The association of instationary outflow with stationary water depths is only possible if the specific distance l_b is considered. For all subsequent derivations it is easier to replace the unknown distance l_b by the doubled amount, which we refer to as characteristic length l_ch.

( ) ( )

¹

( )

¹

l_ch [m] characteristic length

The term ∆h/∆Q(h) corresponds to the gradient of the outflow curve for the given depth h in finite difference form (see Figure 7.18). Therefore it is only necessary to know the stationary outflow curve to determine the reference stretch l_b or the characteristic length l_ch.

In a river stretch with the characteristic length l_ch several simple geometric techniques can be used to determine the volume (see Figure 7.19). If the water surface is linear and similar along the stretch, the volume of the instationary outflow wedge ACF equals the volume of the prism BCFE. Hence the instationary outflow volume V_ADGF can be related directly to the stationary water depth in the middle of the reference stretch

( ) ( )

h [m] water depth in the middle of the reference river stretch.

Subsequently, the results for the middle of a river stretch section with the characteristic length l_ch are listed:

• The one-to-one stationary outflow curve Qstat(h_m) represents the instationary outflow Q_A (Q_A = Q_inst, reservoir outflow).

• The water depth hm indicates the volume S (storage contents).

• Even for instationary flow a direct relation between storage contents and outflow S(QA ) exists.

The water depth h_m in the middle of the selected river stretch constitutes the connection between the outflow Q_A and the storage content S.

Applying ^{QA =QA}

( )

^{hm =Qstat}

( )

^{hm and S =S h}

( )

^{m =A h}

( )

^{m ⋅lch} provides ^{S =S Q}

( )

^{A or}

( )

A A

Q =Q S .

Q_A [m³/s] outflow from river stretch of the length l_ch, S [m³] storage content of the river stretch, length l_ch.

A channel cross-section with the characteristic length l_ch may be regarded as a reservoir.

Instationary flow can be calculated relatively easy with the help of the reservoir retention method. Long flow paths l_Fl can be represented by a series of reservoirs, each with the characteristic length l_ch. The only data required are the representative channel cross-section A(h) and the stationary outflow curve Q_stat(h).

Since the outflow curve is a one-to-one function, h = h(Q_stat), l_ch can be regarded as a function of Q_stat. Looking at Figure 7.20, it becomes obvious that the characteristic length l_ch is not a constant because the slope of the outflow curve is not linear.

( )

^stat

ch ch stat

stat stat

Q dh

l l Q

I dQ

= = ⋅ (7.60)

characteristic length lch (m)

outflow Q_stat (m³/s)

storage constant k (h)

Figure 7.20: Dependency of the characteristic length l_ch and the storage constant on the stationary outflow Q_stat, determination of the relevant flood range, mean values l_ch,m and k_m

In the range of the relevant flood flows, however, the characteristic length l_ch remains almost constant. Therefore a mean characteristic length l_ch,m, that is taken from the upper part of the outflow curve (from Q_min to Q_max) is applied (see Figure 7.20).

l_ch,m [m] mean characteristic length Q_min [m³/s] lower and

Q_max [m³/s] upper boundary of the relevant range of the outflow curve

outflow Q_stat (m³/s)

water depth h (m)

Q_max

Figure 7.21: Stationary outflow curve Q_stat(h), relevant flood range from Q_min to Q_max, subdivision in outflow intervals ∆Q

For practical use the finite difference equation of the characteristic length is more convenient.

The relevant range of the outflow curve is subdivided into m intervals of uniform length ∆Qi , the characteristic length l_ch,i is determined in each interval middle Q_i (see Figure 7.21).

, ,

Another problem is selecting the best number of characteristic river stretch sections. The total length of the examined river stretch l_Fl rarely coincides with an integer multiple of the

characteristic length. L_ch. To solve this, the integer of the quotient is taken and the reservoirs are fitted to this value.

,

n [1] integer number of reservoirs

l_Fl [m] total length of the examined river stretch l^*_ch [m] fitted characteristic length

The cross-sectional area A(h) and the outflow curve Q_stat(h) remain unchanged.

The following steps must be taken:

• determination of a representative channel cross-section A(h),

• determination of the representative stationary outflow Q_stat(h),

• determination of the relevant flood range from Q_min to Q_max,

• subdivision of the flood range into intervals of uniform length ∆Q_i,

• determination of the characteristic length l_ch,i in each interval,

• computation of the mean characteristic length l_ch,m,

• determination of the number of reservoirs n,

• fitting of the characteristic length l_ch to the total length of the river stretch l_Fl,

• determination of the storage content-outflow relation S(Q_A of the reservoirs).

The storage content-outflow relation S(Q_A ) has much more influence on the precision of the results than the characteristic length l_ch or the number of reservoirs n.

Usually attempting to replace the whole river stretch by a number of equal reservoirs is best.

If this is impossible due to inhomogeneities (e. g. if the bed slope or the cross-sectional area change suddenly or tributaries occur), the sections of equal conditions must be considered separately. This being possible is an advantage of this method; additionally it can be fitted easily to the natural conditions. As a result, even natural or artificial reservoirs such as lakes or dams can be taken into consideration.

If continuous outflow measurements for the river stretch are available, usually the calibration based on the outflow hydrographs provides better results than the KM-method.

In document Hydrology 3 (Page 97-107)