This paper presents an evaluation and analysis of resistance parameters:
friction slope, friction velocity and Manning coefficient in unsteady flow.
The methodology to enhance the evaluation of resistance by relations derived
from flow equations is proposed. The main points of the methodology are
(1) to choose a resistance relation with regard to a shape of a channel and
(2) type of wave, (3) to choose an appropriate method to evaluate slope of water
depth, and (4) to assess the uncertainty of result. In addition to a critical analysis of
existing methods, new approaches are presented: formulae for resistance
parameters for a trapezoidal channel, and a translation method instead of
Jones' formula to evaluate the gradient of flow depth. Measurements obtained from
artificial dam-break flood waves in a small lowland watercourse have made it
possible to apply the method and to analyse to what extent resistance
parameters vary in unsteady flow. The study demonstrates that results of
friction slope and friction velocity are more sensitive to applying
simplified formulae than the Manning coefficient (

Resistance is one of the most important factors affecting the flow in open channels. In simple terms it is the effect of water viscosity and the roughness of the channel boundary which result in friction forces that retard the flow. The largest input into the resistance is attributed to water–bed interactions.

Resistance to flow is expressed by friction slope

On the other hand, in engineering practice the resistance is traditionally
characterised by the Manning coefficient (

In unsteady flow additional factors affect flow resistance compared to steady
flow. As

A large variety of methods of bed shear stress and friction velocity evaluation
have been devised in order to study the flow resistance experimentally. The
majority of methods measure bed shear stress indirectly, e.g. using hot wire
and hot film anemometry

In this study we apply formulae derived from flow equations to obtain values
of friction slope,

The site of the experiment in the Olszanka watercourse (upper panel), and the shape of measurement cross sections CS1 and CS2 (lower panel).

The paper is structured as follows: Sect. 2 presents settings of a dam-break
field experiment and measurement data. A methodology of evaluation of friction
slope, friction velocity and

The data originate from an experiment carried out in the Olszanka, which is a
small lowland watercourse in central Poland (see upper panel of Fig.

Trapezoidal cross section of a channel with definitions of symbols used in the text.

Experimental reach of the Olszanka watercourse (courtesy of Jerzy Szkutnicki).

Temporal variability of flow depth

Rating curves of experimental flood waves in the Olszanka watercourse.

In the study, two cross sections, denoted in Fig.

Two data sets are used in this study, denoted as follows: Ol-1, Ol-2. Other
data sets provided qualitatively similar results and therefore, for
simplicity, are not presented herein. The first set was collected in
cross section CS1 and the other in cross section CS2 during the passage of
the same wave on 26 April 1990 at the beginning of the vegetation season when
banks were slightly vegetated (Fig.

The methodology of evaluating resistance to flow from flow equations is
proposed. It comprises four questions that need to be answered to obtain
reliable values of resistance.

What is the shape of the channel – is simplification of the channel geometry applicable?

Is it admissible to apply simplified formulae with regard to the type of wave?

What methods of evaluating input variables, especially the gradient of flow depth, are feasible in the case under study?

What is the uncertainty of the input variables, and which of them are most significant?

In proceeding sections a thorough review of each questioned issue is given. Methods used in the literature are facilitated with critical analysis, and some new approaches are proposed by the authors.

In this study, resistance to flow is evaluated by formulae derived from flow
equations – the momentum conservation equation and the continuity equation.
Here we propose to evaluate resistance to flow for dynamic waves from the
relations derived from the St Venant model for a trapezoidal channel

The friction slope derived analytically from the set of equations is
represented by the following formulae:

Flow equations for rectangular channels or unit width are the most frequently
used mathematical models to derive formulae on resistance. A number of
formulae for friction velocity has been presented in the literature, e.g.:

If the acceleration terms of the momentum balance equation for dynamic waves
(Eq.

Below we provide simplified relations for diffusive waves, which are applied in
this study:

The evaluation of

Paradoxically, kinematic wave approximation is widely applied in cases of
non-kinematic waves where

Comparison between rating curve for flood wave and steady flow with
characteristic points, based on

In order to apply the kinematic wave approximation, the wave celerity must be
evaluated. Celerity can be assessed by the formula derived from the Chezy
equation (Eq.

We propose another approach for evaluation of

Because of the drawbacks of kinematic wave approximation, it is recommended
to evaluate the gradient of the flow depth based on data from two
cross sections

Another drawback of the method is the availability of data. Very often, data
originate from measurements which have been performed for some other purpose.
Consequently, the location of gauging stations and data frequency acquisition
do not meet the requirements of the evaluation of the gradient of flow depth

Due to the linear character of a two-point (backward and forward) difference
quotient, it is not able to represent properly the peak region of a flood
wave. In

The results of resistance evaluation should be given alongside the level of
uncertainty. In the case of unrepeatable experiments,

As presented in Sect.

Temporal variability of gradient of flow depth

The following methods of evaluating

linear approximation denoted as

kinematic wave approximation in the form of the Jones formula (Eq.

wave translation (Eq.

kinematic wave approximation (Eq.

Friction slope

In order to assess to which category of flood wave (dynamic, diffusive or
kinematic) the case under study should be assigned, the terms of the momentum
balance equation are compared. The results are shown in Fig.

Comparison of terms of the momentum balance equation for experimental flood waves in the Olszanka watercourse.

Comparison of friction slope evaluated by formulae for dynamic

Comparison of friction velocity evaluated by formulae for dynamic

Figure

In the case of data set Ol-1

Figure

As can be seen in Fig.

Figure

Values of

Comparison of

Comparison of the relation of

Results for

The variability of resistance in unsteady flow is very often analysed in
terms of flow rate

Proper determination of resistance parameters: friction slope, friction
velocity and Manning coefficient in unsteady flow is very often hampered by
scarcity or high uncertainty of input data. However, when resistance
relations are applied with an awareness of their constraints, and proper
effort is made to minimise the uncertainty of the input data, they are likely to
obtain reliable results. To facilitate the evaluation of resistance
parameters, we have proposed the methodology which provides means to enhance
reliability of results obtained by relations derived from flow equations. The
methodology comprises four questions which help to judge if simplifications
with regard to shape of a channel and type of wave are admissible, to decide
which method of

The paper has demonstrated the application of proposed methodology to experimental data; hence, the detailed conclusions drawn in the study apply to similar cases. The methodology has been applied to assess if the simplified formulae are admissible. The analysis of terms of the momentum balance equation has provided identification of the type of waves. In the first case, Ol-1, which is closer to the dam, the wave has dynamic character along the rising limb and diffusive character along the falling limb. In the second case, Ol-2, the wave is of diffusive character with relatively small difference between water slope and bed slope. Thanks to the uncertainty analysis the reliability of the results of resistance parameters obtained by simplified formulae has been assessed.

The analysis revealed that for

Flood wave phenomena are so complex that it is currently impossible to provide a comprehensive analysis, and the problem of resistance to flow in unsteady non-uniform conditions still poses a challenge. For this reason, more research on resistance in unsteady non-uniform conditions is necessary.

This study has been financed by the National Science Centre, grant no. DEC-2011/01/N/ST10/07395. It was also partially supported within statutory activities (no. 3841/E-41/S/2015) of the Ministry of Science and Higher Education of Poland.

The authors would like to express their appreciation to Jerzy Szkutnicki from the Institute of Meteorology and Water Management for his help in obtaining and interpreting data from the Olszanka watercourse. Edited by: R. Moussa