THE QUASI-STEADY APPROXIMATION

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CHAPTER 13 THE QUASI-STEADY APPROXIMATION
The conservation equations governing 1D
morphodynamics can be summarized as
Conservation of flow mass
Conservation of flow momentum
Conservation bed sediment (sample form using
total bed material load)
For many applications in morphodynamics, however,
it is possible to neglect the time derivatives in
the first two equations, retaining it only in the
Exner equation of conservation of bed sediment.
That is, the flow over the bed can be
approximated as quasi-steady. This result, first
shown by de Vries (1965), is often implicitly
used in morphodynamic calculations without
justification. A demonstration follows.
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NON-DIMENSIONALIZATION USING A REFERENCE STATE
The essence of morphodynamics is in the
interaction between the flow and the bed. The
flow changes the bed, which in turn changes the
flow. Consider a reference mobile-bed
equilibrium state with constant flow velocity Uo
flow depth Ho, bed slope So and total volume bed
material transport rate per unit width qto. For
the sake of simplicity the bed friction
coefficient Cf is assumed to be constant. The
analysis easily generalizes, however, to the case
of varying friction coefficient. The application
of momentum balance to the equilibrium flow
imposes the conditions where ?u is the value of
? at x 0. In a problem of morphodynamic
evolution, the flow and bed can be expected to
deviate from this base state. In general,
then, The following non-dimensionalizations are
introduced
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NON-DIMENSIONALIZATION USING A CHARACTERISTIC
HYDRAULIC TIME SCALE
Note that the non-dimensionalization of time
involves the hydraulic time scale Ho/Uo, which
physically corresponds to the time required for
the flow to move a distance equal to one depth in
the downstream direction. Substituting the
non-dimensional variables into the balance
equations yields the results where
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HOW LARGE IS ?
Bed porosity ?p is typically in the range 0.25
0.45 for beds of noncohesive sediment. The
parameter thus scales the ratio of the volume
transport of solids to the volume transport of
water by a river. For the great majority of
cases of interest this ratio is exceedingly
small, even during floods. A case in point is
the Minnesota River near Mankato, Minnesota, a
medium-sized sand-bed stream. Some sample
calculations follow.
Minnesota River at the Wilmarth Power Plant just
downstream of Mankato, Minnesota, USA. Flow is
from left to right.
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HOW LARGE IS ? contd.
Given below are 13 grain size distributions for
the bed material of the Minnesota River at
Mankato, along with an average of all 13. The
fraction of sediment finer than 0.062 microns in
the bed is negligible such material can be
treated as wash load.
From http//www.usgs.gov/
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HOW LARGE IS ? contd.
During the period 1967-1995 the highest measured
suspended load concentration was 2850 mg/liter,
or C 2850/2.651x10-6 0.001075 the discharge
Q was 340 m3/s, so the volume total suspended
load (bed material load wash load) Qsbw 0.366
m3/s.
From http//www.usgs.gov/
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HOW LARGE IS ? contd.
At a discharge of 340 m3/s, about 79.5 of the
suspended load is wash load, giving a suspended
bed material load Qs of 0.075 m3/s. Estimating
the bedload Qb as about 15 of the total bed
material load, an estimate for the highest value
of Qt of 0.088 m3/s is obtained.
From http//www.usgs.gov/
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HOW LARGE IS ? contd.
Assuming a value of bed porosity ?p of 0.35,
then, an estimate of the very high end of the
value of ? that might be attained by the
Minnesota River near Mankato is The
Minnesota River is by no means atypical of
rivers. The largest values of ? attained in the
great majority of rivers is much less than unity.
The exceptions include streams with slopes so
high that the flows are transitional to debris
flows, streams carrying lahars, or heavily
sediment laden flow from regions recently covered
with volcanic ash, and many streams in the Yellow
River Basin of China.
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NOT ALL FLOWS SATISFY THE CONDITION ? ltlt 1
Double-click on the image to see a debris flow in
Japan. The volume (mass) of sediment carried by
debris flows is of the same order of magnitude as
the volume (mass) of water carried by such flows.
The quasi-steady approximation breaks down for
such flows. Video courtesy Paul Heller.
rte-bookjapandebflow.mpg to run without from
relinking, download to same folder as PowerPoint
presentations.
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HYDRAULIC TIME SCALES
Over short, or hydraulic times scales ?t
Ho/Uo, then, when ? ltlt 1 the governing equations
approximate to That is, the bed can be
treated as unchanging for computations over
hydraulic time scales, even though sediment is
in motion. This is because the condition ? ltlt 1
implies lots of water flows through but very
little sediment, so that the bed does not have
time to change in response to the flow.
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MORPHODYNAMIC TIME SCALES
A dimensionless morphodynamic time t can be
defined as An order-one change ?t corresponds
to a change in dimensioned time i.e. much
longer than the characteristic hydraulic time.
The governing equations thus become That
is, when the time scales of interest are of
morphodynamic scale, the flow can be treated as
quasi-steady even though the bed is evolving, and
thus changing the flow.
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THE DIMENSIONED EQUATIONS WITH THE QUASI-STEADY
APPROXIMATION

• According to the quasi-steady approximation, the
  bed changes so slowly compared to the
  characteristic response time of the flow that the
  flow can be approximated as responding
  immediately. The dimensioned equations thus
  reduce to the following forms
• The quasi-steady approximation greatly simplifies
  morphodynamic calculations. There are, however,
  reasons not to use it. These include
• Cases of rapidly varying hydrographs, when it is
  desired to characterize the
• sediment transport over the entire hydrograph
• Cases when one wishes to capture the effect of a
  flood wave (with a high water surface slope on
  the upstream side of the wave and a low water
  surface slope on the downstream side) on sediment
  transport and
• Cases when the flow makes transitions between
  subcritical and supercritical flow, in which case
  a shock-capturing method capable of automatically
• locating hydraulic jumps is required.

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REFERENCES FOR CHAPTER 13
de Vries, M. 1965. Considerations about
non-steady bed-load transport in open channels.
Proceedings, 11th Congress, International
Association for Hydraulic Research, Leningrad
381-388.