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LECTURE 2: FLUID MECHANICS
Introduction
Conservation of mass and momentum
General types of flow
Laminar vs. turbulent flow
Shear Stress
Reach-average shear stress
Bed roughness and reach average flow velocity
Shear stress partitioning
Local shear stress
Laminar velocity profile
Turbulent velocity profile
Determining u* and zo
Laminar sublayer
Smooth bed
Rough bed
Flow Energy
Forms of stream energy
Bernouilli equation
Navier-Stokes Equation
Derivation
Simplifications
Reynolds number
Froude number
Hydraulic scaling
Geology 412
Spring 2002
Introduction
Water flowing in a channel is subject to two principal forces: gravity and friction. Gravity drives
the flow and friction resists it. The balance between these forces determines the ability of flowing
water to transport and erode sediment.
In addition, we expect mass and momentum to be conserved at cross sections 1, 2, …, n unless
mass or energy are added in between.
Conservation of Mass: Q = A1u1 = A2u2 = ... Anun
Conservation of Momentum:
ρQ1u1 = ρQ2u2 = ... ρQnun
(1)
(2)
(note Q = discharge; A = x-sectional area; u = velocity… so these equations are in volume terms)
We will use these two basic principles to derive the shear stress that acts on the channel bed
(and that transports sediment), the velocity profile in a river, and the equations governing
channel flow.
General Types of Flow
steady:
velocity constant with time
unsteady:
velocity variable with time
uniform:
velocity constant with position
non-uniform:
velocity variable with position
Simple mathematical models of flow in channels can be constructed only if flow is uniform and
steady. Although flow in natural rivers is characteristically non-uniform and unsteady, most
models rely upon the steady uniform flow assumption.
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Laminar vs. Turbulent Flow
Note that water is assumed to be “stuck” to the boundary (the “no-slip” assumption).
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Reach-Average Shear Stress
Natural rivers have local irregularities in bed and bank topography that introduce
significant local convergence and divergence of flow that can impose large local gradients in flow
velocity and shear stress. We use a reach-average view of channels in order to make for a
solvable analytical model.
First, consider the force balance on the volume of water in an entire reach of length L and
slope θ:
Assume that acceleration of flow in the reach is negligible and that the bed is not moving, so
there must be a balance between (1) the gravitational force accelerating the water downstream
and (2) the frictional resistance to flow caused by the boundary, which slows the fluid velocity to
zero at the bed and banks and therefore causes internal deformation of the flow.
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Within the reach, the downstream component of gravitational force is
A L ρ g sin θ
(3)
The total boundary resistance (which is also a force, i.e. = stress · area) for the reach equals
τb L P
(4)
where τb is the average drag force per unit area (shear) on the boundary.
Equating the force moving the flow (3) with the force resisting flow (4) (since we are assuming
no additional energy inputs), we get
τb L P = A L ρ g sin θ
(5)
Rearranging terms and dividing by L yields
τb = (A/P) ρ g sin θ
(6)
If we define the hydraulic radius as R ≡ (A/P) then this simplifies to the standard expression for
the reach-average shear stress
τb = R ρ g sin θ
(7)
Note that for wide channels A/P ≈ D; and for small θ, sin θ ≈ tan θ= S.
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Hence, the reach-average basal shear stress is approximated by the
"depth-slope" product:
τb = ρ g D sin θ
(8)
**The force exerted by flow on the channel bed is proportional to flow depth and slope**
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Bed Roughness & Reach-Averaged Flow Velocity
Prediction of flow velocity is a fundamental problem in fluvial geomorphology that is important
for such problems as flood prediction and the drag force exerted on objects in the flow.
We've now established that the basal shear stress is related to the depth-slope product, but how
do we get at flow velocity?
Chezy (1775) first applied mathematical analysis to the mechanics of uniform flow. He made 2
assumptions:
#1
Exact balance between force driving flow (downslope component of the weight of water)
and the total force of bed resistance (i.e. the same assumption we made in writing
equation 5).
#2
The force resisting the flow per unit bed area (i.e., τb) varies as the square of velocity:
τb = k u2
(9)
where k is a roughness coefficient.
Recall that we can express
Driving force
= Weight of water x sine of bed slope
= ρ g A L sin θ
Resisting force
(10)
= Total bed area x bed shear stress
(11)
= P L τb
Assuming no acceleration [Chezy’s assumption #1 above] then these forces balance and
τb =
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ρ g (A/P) sin θ
(12)
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This is the same as equation 6.
Substituting equation 9 [Chezy’s assumption #2] yields
k u2 = ρ g (A/P) sin θ
(13)
Rearranging in terms of velocity yields
u2 = (ρ g / k) (A/P) sin θ
(14)
Recalling that R ≡ A/P, and sin θ ≈ tan θ = S, then
u2 = (ρ g / k) R S
(15)
u = C (R S)(0.5)
(16)
and hence
where C = (ρ g /k)(0.5)
(17)
Equation 16 is called the Chezy equation and C is called the Chezy Coefficient.
Hence, if both of Chezy’s assumptions are correct, the average velocity in a channel should
increase with the square root of the gradient, the square root of the hydraulic radius (which for
wide shallow channels is equal to the average depth), and a coefficient that reflects the
smoothness of the channel (i.e., the inverse of channel “roughness”).
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Later, empirical investigations (i.e. simultaneous measurements of u, R, and S in experimental
flumes) indicated that C varied slightly with R in any given channel:
C α R(1/6)
…and so a new proportionality was defined,
C = R(1/6) / n
(18)
where “n” is the Manning roughness coefficient (another empirical coefficient).
Substitution of this result into the Chezy equation [eqn. 16] produced the famous Manning
Equation:
u = k1 R(2/3) S(1/2) / n
(19)
Manning’s n is a roughness coefficient that depends on channel margin irregularity and the grain
size of the bed material. The scaling of n has been chosen so that constant k1 = 1 in SI units and
1.49 in English units (fps). This has been known to cause confusion.
Manning’s n reflects the net effect of all the factors causing flow resistance in a fluid of a given
viscosity (because of the temperature effect on viscosity, a channel’s n varies slightly throughout
the year).
The third common roughness equation is the Darcy-Weisbach equation for frictional losses in
circular pipes, which can be modified for open channel flow:
ff = 8 g R S / u2
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(20)
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We will omit the derivation for this equation, but it too has its advocates because the DarcyWeisbach friction factor has the advantage of being dimensionless, and hence the units don't
matter.
The three common roughness coefficients are all interrelated:
C = R(1/6) / n
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ff = 8 g n2 / R(1/3)
2-9
ff = 8 g /C2
(21)
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Shear Stress Partitioning
The force available for transporting sediment is that component of the basal shear stress that is
not dissipated by flow roughness, which can be viewed as the sum of:
1.
Grain (or "skin") resistance (or "roughness") due to the presence of small, distributed
irregularities such as bed-forming material.
2.
Form resistance due to the larger-scale internal deformation in the flow field imposed by
channel bed irregularities such as bedforms (e.g., dunes, bars, pools, etc...) and by
variations in the plan form of the river (e.g., meanders).
3.
Spill resistance due to surface waves generated by large obstacles protruding from banks,
steps in the channel bed profile, or other obstacles such as logs and boulders.
The reach average basal shear stress (τb) is often considered to be composed of linearly additive
components of shear stress attributable to these different aspects of flow resistance:
τb = τg + τbf + τs + τ'
(22)
where:
τg
is the grain roughness,
τbf
is the roughness due to bedforms,
τs
is the net effect of other sources of roughness (e.g., logs).
…and so τ' is the effective shear stress available for sediment transport.
Rearranging (22) yields
τ' = τb – (τg + τbf + τs)
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(23)
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Consider τ' as the force left over to move stuff after accounting for the forms of roughness that
impede flow through the channel.
Bedform roughness (τbf ) can account for 10 – 70% of the total roughness in channels with welldeveloped, macro-scale bedforms. In forest landscapes, roughness attributable to in-channel
wood debris (τs) also can account for a substantial portion of the reach-average basal shear stress.
Accounting for these various forms of roughness is a major challenge for predicting flow velocity
and sediment transport, and it is done in 3 typical ways:
1.
analogy — tables, books, pictures
2.
theory — simplify and predict
3.
measurement — field determinations of flow parameters to back-calculate roughness.
You will have an opportunity to practice all three on the first field trip.
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Local Shear Stress (The View at a Point Within a Channel Reach)
Imagine any point within the channel at which the flow can be reasonably viewed as onedimensional and parallel to the bed (three-dimensional complexities add a lot of mathematics
which is ignored in the following).
H
z
θ
The shear stress on any surface at height z above the bed is caused by the downslope
gravitational stress of the water above the plane - i.e., by the downslope component of the weight
of the fluid between z and the water surface (at height H).
Hence, the shear stress at any point within the fluid will be given by :
τ= ρ g (H–z) sin θ
(24)
Equation (24) indicates that the shear stress decreases linearly with height above the bed.
surface
zs
H
bottom
τ
τb
Note also that for the case of z = 0 (i.e., at the channel bed), τ = τb and so equation (24)
reduces to :
τb = ρ g D sin θ
(25)
which we've seen before as equations (7) and (12).
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Laminar Velocity Profile
Water is a viscous fluid that cannot resist a shear stress, however small. It deforms, or strains.
Newton found by experiment that for laminar flow,
τ = μ du/dz
(26)
where τ is shear stress; μ = viscosity; and du/dz is the strain rate.
Or:
strain rate = shear stress / viscosity [du/dz = τ / μ ]
Or:
“The more your push, the faster it goes."
Combining (26) with (24) above [shear stress distribution in the flow]
τ = μ du/dz = ρ g (H–z) sin θ
(27)
Rearranging yields:
du = (ρ g sin θ / μ)H dz – (ρ g sin θ / μ)z dz
(28)
u = (ρ g sin θ / μ) (Hz) – (ρ g sin θ / μ) (z2 / 2) + C
(29)
Integrating:
Combining terms and using the boundary condition that u = 0 when z = 0 [which inspection of
(29) shows implies that C = 0] yields:
u = (ρ g sin θ / μ) [Hz – (z2 / 2)]
(30)
This equation defines the parabolic velocity
profile of laminar flow, which describes the velocity in
many debris flows or very close to the bed of a river
(“the laminar sublayer”).
Farther from the bed in most rivers, the flow paths
of water parcels in the turbulent flow become erratic
and develop into eddies, in which velocity
components in x, y, and z directions fluctuate
randomly about a mean value.
U (average, for a given depth)
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Turbulent Velocity Profile
Turbulent flow mixing between adjacent layers in the flow involves transfer of momentum via
large scale eddies, which impart an extra "eddy viscosity" term (ε) that can be considered
analogous to momentum transfer by conventional viscosity:
τ = (μ + ε) (du/dz) ≈ ε (du/dz)
(31)
This works because typically ε >> μ and hence turbulent flow is slower than laminar flow at the
same shear stress. This is because the drag from the bed is transferred more efficiently into the
body of the flow by eddies than by viscosity alone.
It is extremely difficult to determine the eddy viscosity, but Prandtl proposed that the eddies
would have a length scale (a distance across which they could exchange momentum between
layers in a unit of time) that was proportional to the distance away from the solid/fluid boundary
-- eddying would be suppressed near the boundary. He also proposed that ε depended on the
velocity gradient (du/dz). Thus he developed an expression for the eddy viscosity
ε = ρ l2 (du/dz)
(32)
where ρ is the density of water and “l” is Prandtl's mixing length, which depends on proximity to
the boundary and was experimentally determined as
l=κz
(33)
where κ = 0.4
Equation (33) can be substituted back into (32) and then (31) to yield
τ = κ2 z2 ρ (du/dz)2
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(34)
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Prandtl then introduced the concept of the "shear velocity" (u*), which is not really a velocity
but has the dimensions of velocity [i.e. L / t]. It is assumed to be constant near the bed, where τ
was also assumed to be constant and equal to τb:
u* = (τb / ρ)0.5 = (gHS)0.5
(35)
For τ = τb, incorporating (35) back into (34) yields
u* = κ z (du/dz)
(36)
du = (u*/κ) (dz / z)
(37)
Rearranging (36) yields
Integrating and rearranging terms yields
u = (u* / κ) ln z + C
(38)
If we impose the boundary condition that u = zero at some elevation z0, just above the bed,
then:
0 = (u* / κ) ln z0 + C
(39)
C = –(u* / κ) ln z0
(40)
u = (u* / κ) ln z – (u* / κ) ln z0
(41)
and therefore
Hence, (38) becomes
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which can be simplified to
u = (u* / κ) ln (z/z0)
(42)
This is the "Law of the Wall" (i.e., the equation for turbulent velocity distribution away from, but
“close to,” a fixed boundary such that τ ≈ τb).
surface
ln (z)
lnZ
Z0
bottom
Uu
z = zo
The "Law of the Wall" predicts a logarithmic velocity profile that begins at a roughness length
scale that defines the height above the bed of z0. Below this height flow is must be assumed to be
laminar, because it is indeterminate under our turbulent assumptions (since u = 0 at z = z0).
Note that κ in equations 33–42 is called von Karman's constant (and = 0.4).
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Reiterating:
The solution for the velocity profile in a turbulent river assumes:
1
Newton's viscous flow law applies, as modified in (31) to include an eddy viscosity.
2
l = κz in the neighborhood of the boundary, i.e. turbulent mixing is scaled by distance to
the bed.
3
τ = τb is constant “close” to the boundary.
Farther from the boundary, τ ≠ τb, and perhaps at such points in the interior of the fluid the eddy
viscosity will depend not on the local distance from the bed (z) but rather the on total flow depth
(H). If so, it will be constant across this “interior flow.” Mathematically, this is equivalent to
equation (30), i.e. a constant “viscosity”’ (only in this case it’s an eddy viscosity). As a result, the
velocity profile in the interior of the flow will also be parabolic (see equation 30), although with a
different viscosity than in the laminar sublayer.
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Determining u* and z0
= u*2,
the slope of the velocity profile on a semi-log plot can be used to measure the local shear stress,
Since the slope of the velocity profile is a measure of u*, the shear velocity, and since
particularly near the channel bed, either over bedforms, or (if the velocity profile can be defined
sufficiently close to the bed) over the grains themselves.
To obtain u* and z0 in equation (42), measure u at various heights, z, above the bed. If you take
the natural logarithm of the z values, then if the points conform to (42) they will plot as a
straight line (where the x-axis is velocity and the y-axis is ln z) because (42) would be written as
u = (u*/κ) ( ln z – ln z0 )
(43)
Hence u* can be calculated from either the best-fit line through paired values of u and ln z data
or by reading pairs of data and using the equation for the slope of a line.
If you plot the logarithm of flow depth on the y-axis and velocity on the x-axis, then the slope of
the line is given by:
κ/u* = (ln z1 – ln z2 ) / (u1 – u2)
(44)
Hence, if you take a linear regression of ln z (the natural logarithm of the flow depth at which
each velocity measurement was made) versus the flow velocity (u) then in the slope-intercept
form of the expression (y = mx + b), the slope of that line (m) is given by κ/u* and the intercept
of that line (b) is equal to ln z0.
So
z0 = eb
And you can calculate u* as:
u* = κ / m
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Because the theory tries to specify conditions only close to the solid boundary it is strictly a
reasonable approximation only close to the boundary and has therefore become known as "the
Law of the Wall".
Farther away from the bed, the mixing length becomes constant at (an empirically determined)
fraction of the total depth and the velocity profile becomes parabolic above that depth. Log and
parabolic profiles predict the same velocity at 0.2H, which is the presumed level of this
“transition.” However, the difference between the computed logarithmic and upper parabolic
profiles in most streams is negligible, and so for many applications a logarithmic profile can be
assumed throughout.
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Laminar Sublayer
Very close to the bed, velocity is low and turbulence is suppressed, so the flow is laminar.
Above this "laminar sublayer" (also sometimes called the “viscous sublayer”), the turbulent
velocity profile with its apparent z0 begins.
The thickness of the sublayer (ζv) depends on the near-bed shear velocity. By
dimensional analysis it should have a thickness proportional to (μ/ρu*); by experiment, the
generally accepted equation for the sublayer thickness is
ζv = 11.6 ν / u*
(45)
where ν is the kinematic viscosity (μ/ρ) [Recall that u* = (τb / ρ)0.5]
[Note that ν = 1 x 10–2 cm2/s (1 centistoke) or 1 x 10–6 m2/s at 20°C]
ks
So, what is the scale of ζv for flow in a typical gravel bed river with a depth of 1 m and sin θ =
0.005? (about 0.05 mm, but work it out yourself!)
What is the scale of ζv for flow in a typical gravel bed river with a depth of 2 m and sin θ =
0.035? [high estimates] (≈ 0.01 mm)
What is the scale of ζv for flow in a typical gravel bed river with a depth of 0.5 m and sin θ =
0.001? [low estimates] ] (≈ 0.2 mm)
Hence, the length scale of ζv is about the diameter of silt to fine sand grains.
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Smooth Bed
If the laminar sublayer is much thicker than the size of roughness elements on the bed
(ks), the surface is considered “smooth.” What size of bed material would allow hydraulically
smooth flow where the turbulence doesn't interact with the bed roughness? We can already
expect that ks must be “much” less than 11.6 ν / u*.
ks
Note that we can define a dimensionless ratio of the laminar sublayer thickness to the roughness
elements on the bed. This has been termed the “Roughness Reynolds number,” and for
dimensional homogeneity (and linear dependence of ζv on ν and u ):
*
Re* = ks u* / ν
(46)
From (45), we know that this ratio must be “much” less than 11.6 (because ks must be “much”
less than ζv for hydraulically smooth flow to occur), but only experiments can determine just how
much less. The answer is 3. Thus, for hydraulically smooth flow,
3 ≥ ks u* / ν
(47)
For hydraulically smooth flow, measured velocity profiles in the overrunning turbulent flow
indicate an apparent z0 of
Combining (45) and (48) yields:
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z0 ≈ ζv / 100
(48)
z0 ≈ ν / (9 u*) (i.e. very small!)
(49)
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Rough Bed
If the bed roughness elements are large relative to v (i.e., > sand or fine gravel), then
the laminar sublayer will rise and fall over the protuberances, and the grains will begin
contributing addition form drag in addition to ordinary surface friction:
ks
Consequently, turbulence interacts directly with the roughness elements causing z0 to be scaled
by their size. We know that ks must be “much” greater than ζv and thus that
Re* must be “much” greater than 11.6, but once again experiments were required to determine
just how much. Nikuradse's experiments for such "hydraulically rough flow" showed that it
occurred when:
ks u* / ν ≥ 100
(50)
He also anticipated that the value of z0 would depend on ks; by further experiment,
z0 = ks / 30
(51)
Substitution of (51) into the "Law of the Wall" yields
u = (u* / κ) ln (30 z/ks)
(52)
Field measurements have shown D84 to provide a reasonable measure of ks, although Whiting
and Dietrich (1991) reported field-measured z0 values that were about 3 times larger than
predicted by equation 51.
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Flow Energy
Precipitation over a landscape results in downslope movement of water, causing erosion
and energy expenditure that forms and maintains channels. The frequency and magnitude of
precipitation and the topographic relief onto which it falls provide the source of this potential
energy.
For the simple case of spatially-uniform rainfall, the potential energy (Ep) in a catchment is equal
to the integral of the product of water mass (m), gravitational acceleration (g), and elevation (z)
Ep
=
∫ m g dz
(53)
Initially, the total energy of the system (E) consists of potential energy (mgz).
Downslope movement of water converts this potential energy into kinetic energy (mu2/ 2),
pressure energy (mgD), and energy dissipated by friction (F) and turbulence. Conservation of
energy implies that ∆E = 0 and hence this dissipative system is charcterized by
∆E = 0 = ∆(mgz) + ∆(mu2/ 2) + ∆(mgD) – F
(54)
where u and D are respectively the flow velocity and depth.
The loss of potential energy is compensated by increased flow velocity, increased flow depth,
and/or greater frictional energy dissipation. Thus,
F = ∆(mgz) + ∆(mu2/ 2) + ∆(mgD)
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(55)
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Combining the bed elevation (z) and the flow depth (D) into a water surface elevation (H)
allows recasting (55) as
F = ∆(mgH) + ∆(mu2/ 2)
(56)
Assuming that change in the downstream flow velocity is small [i.e., ∆(mu2/ 2) ≈ 0], then the
rate of frictional energy dissipation is related to the fall in the water surface per unit channel
length (L):
F/L = mg ∆H/L
(57)
The frictional energy dissipation per unit channel length effectively scales the channel roughness
(R). Noting that ∆H/L is the water surface slope (S), implies that R α S.
In general, changes in slope dominate flow depth changes (Leopold et al., 1964). Since channels
tend to be steep in their headwaters and decrease in slope downstream, this implies that channel
roughness generally decreases downstream.
This leads to the rather counter-intuitive result that steep headwater channels flow slower than
their lowland counterparts.
For many years geologists simply asserted that steep headwater channels obviously flowed faster
than their lowland counterparts.
In 1953 Luna Leopold showed that this conventional wisdom was incorrect by having the
audacity to actually go out and measure stream velocity at many points down a channel network.
This effect is due to the greater roughness of steeper channels -- low gradient rivers can be
deceptively fast!
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Bernoulli Equation
The Bernoulli equation describes the interrelation of stream slope, water surface depth, and flow
velocity based on conservation of energy.
Total energy of a unit volume of flow:
potential energy:
ρgh
pressure energy:
ρ g d cos θ
kinetic energy:
ρu2/ 2
E = ρ g h + ρ g d cos θ + ρu2/ 2
(58)
For small slopes d ≈ d cos θ and thus (58) can be re-expressed as
E = ρ g [ h + d + (u2/ 2g)]
(59)
The term in parentheses is the total head (H) and flow is driven from high to low head. Note
that “H” is now a distance above the datum, not the total flow depth as before:
H = h + d + (u2/ 2g)
(60)
This is the Bernoulli equation which describes conservation of energy from reach to reach.
Consider two reaches (designated with subscript 1 and 2):
The head loss between the reaches (∆H) will be equal to H1 – H2 and hence
h1 + d1 + (u12/ 2g) = h2 + d2 + (u22/ 2g) + ∆H
(61)
Note that the energy, water surface, and bed slopes are not necessarily parallel.
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Navier-Stokes Equation—Derivation of the full equations of fluid motion
Up to this point, we have made implicit assumptions about the flow, particularly its steady and
uniform nature. It is instructive, however, to reconstruct our derivations by starting with the full
equations of fluid motion, in order to remember what we ultimately must leave out and to
understand where some of our most useful flow parameters actually come from.
The basic principle is Newton’s Second Law:
F=ma
(62)
This can be stated in words that the rate of change of momentum of a body is equal to the
force(s) acting on that body (or particle, or infinitesimal element of material, or whatever).
Recall that “momentum” is equal to mass (m) times velocity (u), and acceleration (a) is the first
derivative of velocity with respect to time (i.e. the rate of change). Because we do not expect
mass to change with time,
d(mu) /dt = m du/dt = m a
(63)
This becomes complicated only because we need to address both “body forces” (gravity is the
most common of these) and “surface forces” (also called “tractions”), and because if we are being
complete then we must deal with them in all 3 dimensions.
The notation for Newton’s second law in 3 dimensions, with body and surface forces called out
separately, expressed per unit volume, is:
d
⋅ [u˜ ⋅ ρ ] = [− g˜ ⋅ ρ ] + [∇ ⋅ τ˜ ]
dt
(64)
This is Cauchy’s first law, and it applies to any material (since we have only made the assumption
that it behaves in accord with Newton’s second law). It says, in tensor notation (i.e. vectors in 3
dimensions, indicated by the “~” symbol over the 3-D variables), that the change in momentum
(per unit volume) equals the sum of the body force (gravity, only—no magnetic fields allowed!)
and the surface tractions (τ—more about them later).
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We can expand and rearrange this equation slightly:
ρ
du˜
= −[∇ ⋅ p] + [∇ ⋅ τ˜ ]− [ρ ⋅ g˜ ]
dt
(65)
We have separated the surface forces into those that apply a shear (τ) and those that act
isotropically (p), which we normally call the “pressure.” It is defined as:
p≡
τ 11 + τ 22 + τ 33
3
(66)
where τij is the notation whereby the force in question is acting on the face perpendicular to the
“ith” axis and is applied in the direction parallel to the “jth” axis.
With equation (66), we are stuck until the non-isotropic (also called the “deviatoric”) part can
be expanded. To do this, we need a constitutive equation that relates strain (deformation, or
movement) of the material to the applied stress (which, by definition, is a force per unit area).
This requires experimentation. Fortunately, there is a large class of common materials that
behave rather simply: their strain gradient is proportional to the applied shear stress. In (3-D)
tensor notation, this can be written as:
du j
dx i
∝ τ ij
(67)
The proportionality constant? For these materials, called “Newtonian fluids,” that constant
(which will vary for different substances, but which is the same value in any direction and under
any applied stress regime), is called the viscosity (μ). We’ve done this already, but we came at it
then with a less explicit set of simplifying assumptions (see equation 26). If we add the additional
requirements that the material is incompressible and isotropic, Cauchy’s first law (equation 65)
becomes:
ρ
du˜
= −[∇ ⋅ p] + [μ∇ 2u˜ ]− [ρ ⋅ g˜ ]
dt
(68)
This is the Navier-Stokes equation for incompressible, isotropic Newtonian fluids.
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Now what? This still cannot be solved analytically. So let’s radically simplify things:
1. Steady flow: so all the ∂/∂t terms go to zero
2. 2-D flow: so all the ∂/∂y terms go to zero (no cross-stream variations)
3. Uniform flow: so all the ∂/∂x terms go to zero, except the downstream pressure gradient ∂p/∂x
(otherwise this is just an exercise in statics!).
These simplifications, applied to equation (68), yield two equations (for the x and z directions):
∂p/∂x = ∂ τzx/∂z
(69)
∂p/∂z = – ρg
(70)
p = –ρgz + C
(71)
Integrating (70) yields:
and since the pressure equals 0 (atmospheric) at the water surface (where z = H), we can define
h as the distance down from the level z = H:
p = ρgh
(72)
ρg ∂h/∂x = ∂ τzx/∂z
(73)
This is the “hydrostatic equation.”
Substitute (72) into (69):
and integrate this equation with respect to z:
τzx = ρg(H–z) (∂h/∂x)
(73)
or:
τzx = ρgh tan θ
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(74)
Spring 2006
This one is pretty familiar, too! (Note that if h is measured perpendicular to the bed instead of
vertically, the equation is τzx = ρgh sin θ as in equation (25).)
Finally, we can add our Newtonian constitutive relationship (equation 67) for our 2-D flow:
τzx = μ (∂u/∂z)
(75)
…and solve for u:
u = (ρ g sin θ / μ) [Hz – (z2/ 2)]
(76)
This is also equation (30) from our earlier discussion.
Note that this holds strictly for steady, 2-dimensional flow. This rules out turbulence! Even the
simple hydrostatic equation was “built” from these same assumptions, and so strictly speaking it
too applies only for non-turbulent conditions. We can evaluate whether we need to worry about
turbulence, and we can also figure out what to do about it, using two different approaches.
First, we can just “pretend” that it doesn’t matter and make some experimental measurements.
From these, we find that the basic equations derived from the Navier-Stokes equation (i.e.
equations (72) and (74)) work pretty well, virtually all of the time. So we’ll continue to use
them.
For the velocity distribution (76), however, results are not so friendly. We already found that
where turbulence is “important”,
τ = (μ + ε) (du/dz) ≈ ε (du/dz)
(31)
and the form of the “eddy viscosity” (ε) leads to a logarithmic (as opposed to a parabolic)
velocity profile wherever that viscosity depends on height above the bed.
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Reynolds Number
To decide if turbulent flow is likely, the dimensionless Reynolds number was defined. The
Reynolds number distinguishes laminar and turbulent flow on the basis of the ratio between
inertial and viscous forces. It is named after the Irish engineer Osborne Reynolds (1842–1912)
who first showed that the transition from laminar to turbulent flow generally takes place at a
critical value of the Reynolds number (Re ).
Inertial “Force”:
Viscous “Force”:
ρ u D = density x velocity x flow depth
μ = viscosity = defines relation between applied stress and the strain rate,
or the resistance of the material to deformation.
Re = ρ u D / μ
Laminar flow:
(77)
Re < 500
Viscous forces large relative to inertial forces, as evidenced by little vertical mixing.
Transitional flow:
500 < Re < 2000
Fully turbulent flow:
Re > 2000
Inertial forces >> viscous forces, as evidenced by chaotic streamlines.
Velocity components of turbulent flow consist at any point of a time-average mean velocity (i.e.,
üx) and flucutating velocity components (i.e., ux')
ux = üx + ux'
uy = üy + uy'
(78)
uz = üz + uz'
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The mean values of ux', uy', and uz' are zero, but their standard deviations are non-zero and scale
the intensity of turbulence (It):
It = ( [(ux'2 + uy'2 + uz'2) / 3 ]0.5 ) / üx
(79)
Froude Number
The Froude number, Fr, named for the English engineer William Froude (1810–1879) is
important because it is the ratio between the velocity of streamflow (u) and that of a shallow
gravity wave [(gd)0.5], or the ratio of inertial forces to gravity forces, as simplified as follows:
Fr = u / (gd)0.5
(80)
Think of the Froude number as a measure of whether flow can outrun its own wake.
Subcritical flow Fr < 1
Flow is tranquil and the wave speed exceeds the flow velocity so that ripples on the water surface
are able to travel upstream.
Supercritical flow
Fr > 1
Flow is rapid and gravity waves cannot migrate upstream. Surface waves are unstable and may
break, which greatly increases resistance to flow.
The Reynolds and Froude numbers are independent of the scale of the river and hence provide
dimensionless ways to characterize flow. They also have distinct physical manifestations in the
behavior of flow in a channel.
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Note that laminar flow requires depth and velocity combinations too small for most channel
flows, but would be more common for thin flows (i.e., sheetwash) on hillslopes.
Turbulent flow (Re > 2000) is virtually inevitable in open channel flow. In contrast, both
subcritical (Fr < 1) and supercritical (Fr > 1) flows are common in natural rivers.
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Hydraulic Scaling
Rivers are hard to control—so much of fluvial geomorphology has relied on flume experiments.
It is important in making analogies between hydraulic models and real rivers to have similar
Reynolds and Froude numbers.
Froude Scaling: Equal Froude numbers implies that
[u / (gd)0.5] model = [u / (gd) 0.5] real
If g cannot change, then this implies that
(um / ur)2 = (dm / dr)
(81)
(82)
Hence, if the depth ratio is 1/100, then the velocity ratio must be 1/10.
Reynolds Scaling: Equal Reynolds numbers implies that
(ρud / μ )model = (ρud / μ )real
(83)
or by rearranging
(dmum / drur) = (μmρr / μrρm)
(84)
If also subject to Froude scaling we can substitute (82) into (84) to yield
(dm / dr)3/2 = (μmρr / μrρm)
(85)
For a length scale of 1/100 (typical for flumes) rearranging (85) yields
(μm / μr) = .001 (ρm / ρr)
(86)
Almost all common liquids have densities close to that of water—but we need something 1000
times less viscous! Hence, it is impossible to achieve both Reynolds and Froude scaling in
experimental work.
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