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 (1)

Conservation of Momentum: ρQ1u1 = ρQ2u2 = ... ρQnun (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.

ESS 426 2-1 Spring 2006

Laminar vs. Turbulent Flow

Note that water is assumed to be “stuck” to the boundary (the “no-slip” assumption).

ESS 426 2-2 Spring 2006

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.

ESS 426 2-3 Spring 2006

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.

ESS 426 2-4 Spring 2006

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**

ESS 426 2-5 Spring 2006

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 (10)

Resisting force = Total bed area x bed shear stress

= P L τb (11)

Assuming no acceleration [Chezy’s assumption #1 above] then these forces balance and

τb = ρ g (A/P) sin θ (12)

ESS 426 2-6 Spring 2006

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)

and hence

u = C (R S)(0.5) (16)

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”).

ESS 426 2-7 Spring 2006

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 (20)

ESS 426 2-8 Spring 2006

We will omit the derivation for this equation, but it too has its advocates because the Darcy-

Weisbach 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 ff = 8 g n2 / R(1/3) ff = 8 g /C2 (21)

ESS 426 2-9 Spring 2006

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) (23)

ESS 426 2-10 Spring 2006

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 well-

developed, 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.

ESS 426 2-11 Spring 2006

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 one-

dimensional and parallel to the bed (three-dimensional complexities add a lot of mathematics

which is ignored in the following).

θ

H

z

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.

τ

H

surface

τb

zs

bottom

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).

ESS 426 2-12 Spring 2006

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)

Integrating:

u = (ρ g sin θ / μ) (Hz) – (ρ g sin θ / μ) (z2 / 2) + C (29)

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”).

U (average, for a given depth)

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.

ESS 426 2-13 Spring 2006

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 = κ z (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 (34)

ESS 426 2-14 Spring 2006

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)

Rearranging (36) yields

du = (u*/κ) (dz / z) (37)

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)

and therefore

C = –(u* / κ) ln z0 (40)

Hence, (38) becomes

u = (u* / κ) ln z – (u* / κ) ln z0 (41)

ESS 426 2-15 Spring 2006

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).

z = zoU

lnZ

surface

Z0

bottom

ln (z)

u

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).

ESS 426 2-16 Spring 2006

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.

ESS 426 2-17 Spring 2006

Determining u* and z0

Since the slope of the velocity profile is a measure of u*, the shear velocity, and since � = �u*2,

the slope of the velocity profile on a semi-log plot can be used to measure the local shear stress,

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

ESS 426 2-18 Spring 2006

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.

ESS 426 2-19 Spring 2006

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.

ESS 426 2-20 Spring 2006

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

z0 ≈ ζv / 100 (48)

Combining (45) and (48) yields: z0 ≈ ν / (9 u*) (i.e. very small!) (49)

ESS 426 2-21 Spring 2006

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.

ESS 426 2-22 Spring 2006

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) (55)

ESS 426 2-23 Spring 2006

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!

ESS 426 2-24 Spring 2006

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: ρ g h

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.

ESS 426 2-25 Spring 2006

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 = m a (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:

ddt

⋅ ˜ u ⋅ ρ[ ]= − ˜ g ⋅ ρ[ ]+ ∇ ⋅ ˜ τ [ ] (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).

ESS 426 2-26 Spring 2006

We can expand and rearrange this equation slightly:

ρd˜ u dt

= − ∇⋅ p[ ]+ ∇ ⋅ ˜ τ [ ]− ρ ⋅ ˜ g [ ] (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 ≡τ11 + τ22 + τ33

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:

duj

dxi

∝ 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:

ρd˜ u dt

= − ∇⋅ p[ ]+ μ∇2 ˜ u [ ]− ρ⋅ ˜ g [ ] (68)

This is the Navier-Stokes equation for incompressible, isotropic Newtonian fluids.

ESS 426 2-27 Spring 2006

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)

Integrating (70) yields:

p = –ρgz + C (71)

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)

This is the “hydrostatic equation.”

Substitute (72) into (69):

ρg ∂h/∂x = ∂ τzx/∂z (73)

and integrate this equation with respect to z:

τzx = ρg(H–z) (∂h/∂x) (73)

or:

τzx = ρgh tan θ (74)

ESS 426 2-28 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.

ESS 426 2-29 Spring 2006

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”: ρ u D = density x velocity x flow depth

Viscous “Force”: μ = viscosity = defines relation between applied stress and the strain rate,

or the resistance of the material to deformation.

Re = ρ u D / μ (77)

Laminar flow: 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'

ESS 426 2-30 Spring 2006

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.

ESS 426 2-31 Spring 2006

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.

ESS 426 2-32 Spring 2006

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 (81) If g cannot change, then this implies that (um / ur)2 = (dm / dr) (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.

ESS 426 2-33 Spring 2006