LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW

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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The basic equations of incompressible Newtonian
fluid mechanics are the incompressible forms of
the Navier-Stokes equations and the continuity
equation
These equations specify four equations
(continuity is a scalar equation, Navier-Stokes
is a vector equation) in four unknowns ui (i
1..3) and p.
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The physical meaning of the terms in the
Navier-Stokes equations can be interpreted as
follows. Multiplying by ? and using continuity,
the equations can be rewritten as
A B
C D E
Term A time rate of change of momentum Term B
pressure force Term C net convective inflow
rate of momentum inertial force Term D
viscous force net diffusive inflow rate of
momentum Term E gravitational force
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
We make the transformations (u1, u2, u3) (u, v,
w) and (g1, g2, g3) (gx, gy, gz). Expanding
out the equations we then obtain the following
forms for the Navier-Stokes equations
and the following form for continuity
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The simplest flow we can consider is constant
rectilinear flow. For example, consider a flow
with constant velocity U in the x direction and
vanishing velocity in the other directions, i.e.
(u, v, w) (U, 0, 0). This flow is an exact
solution of the Navier-Stokes equations and
continuity.
Thus for any constant rectilinear flow, all that
needs to be satisfied is the hydrostatic pressure
distribution (even though there is flow)
or
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW

• For plane Couette flow we make the following
  assumptions
• the flow is steady (?/?t 0) and directed in
  the x direction, so that the only velocity
  component that is nonzero is u (v w 0)
• the flow is uniform in the x direction and the z
  direction (out of the page), so that ?/?x ?/?z
  0
• the z direction is upward vertical
• the plate at y 0 is fixed and
• the plate at y H is moving with constant speed
  U
• For such a flow the only component of the viscous
  stress tensor is

H
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
That is, the components of the viscous stress
tensor are
Here we abbreviate
?
moving with velocity U
u
fluid
H
y
x
fixed
?
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Thus u u(y) only, and v w 0. This result
automatically satisfies continuity
Momentum balance in the x, y and z directions (z
is upward vertical)
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Momentum balance in the z direction (out of the
page)
That is, the pressure distribution is
hydrostatic. Recall that the general relation
for a pressure distribution ph obeying the
hydrostatic relation is
H
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Momentum balance in the x (streamwise) direction
The no-slip boundary conditions of a viscous
fluid apply the tangential component of fluid
velocity at a boundary the velocity of the
boundary (fluid sticks to boundary)
H
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Integrate once
Thus the shear stress ? must be constant on the
domain.
Integrate again
Apply the boundary conditions
to obtain C2 0, C1
U/H and thus
H
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW

• For open-channel flow in a wide channel we make
  the following assumptions
• the channel has streamwise slope angle ?
• x denotes a streamwise (not horizontal)
  coordinate, z denotes an upward normal (not
  vertical) coordinate and y denotes a
  cross- stream horizontal coordinate
• the flow is steady (?/?t 0) and directed in
  the x direction, so that the only velocity
  component that is nonzero is u (v w 0)
• the flow is uniform in the x direction and the y
  direction (out of the page), so that ?/?x ?/?y
  0
• the bottom of the channel at z 0 is fixed
• there is no applied stress at the free surface
  where z H.

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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The channel width is denoted as B. It is assumed
that the channel is sufficiently wide (B/H ltlt 1)
so that sidewall effects can be ignored. Thus
streamwise velocity u is a function of upward
normal distance z alone, i.e. u u(z).
H
B
The vector of gravitational acceleration is (gx,
gy, gz) (gsin?, 0, -gcos?)
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Continuity is satisfied if u u(z) and v w 0.
The equations of conservation of streamwise and
upward normal momentum reduce to
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The equations thus reduce to
Since
The first equation can thus be rewritten as
where ? is an abbreviation for ?13 ?31.
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Assuming that a) pressure is given in gage
pressure (i.e. relative to atmospheric pressure)
and there is no wind blowing at the liquid
surface, the boundary conditions on
are
viscous fluid sticks to immobile bed
no applied shear stress as free surface
gage pressure at free surface 0 (surface
pressure atmospheric)
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Now the condition
states that the hydrostatic relation prevails
perpendicular to the streamlines (which are in
the x direction). Integrating the relation with
the aid of the boundary condition
yields a pressure distribution that varys
linearly in z
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The equation
subject to
similarly yields a linear distribution for shear
stress ? in the z direction
Note that the bed shear stress ?b at z 0 is
given as
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Thus
subject to
Integrates to give the following parabolic
profile for u in z
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The maximum velocity Us is reached at the free
surface, where z H and ? 1)
Thus
Depth-averaged flow velocity U is given as
Thus
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
A dimensionless bed friction coefficient Cf can
be defined as
Here Cf f/8 where f denotes the Darcy-Weisbach
friction coefficient. Between the above relation
and the relations below
it can be shown that
Here Re denotes the dimensionless Reynolds No. of
the flow, which scales the ratio of inertial
forces to viscous forces.
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Now suppose that there is a wind blowing upstream
at the free surface, exerting shear stress ?w in
the x direction. The governing equations of
the free surface flow remain the same as in Slide
15, but one of the boundary conditions changes to
The corresponding solution to the problem is
where r is the dimensionless ratio of the wind
shear stress pushing the flow upstream to the
force of gravity per unit bed area pulling the
flow downstream
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LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The solution for velocity with the case of wind
can be rewritten as
where und is a dimensionless velocity equal to
2u?/(gsin?H2).
A plot is given below of und versus ? for the
cases r 0. 0.25, 0.5, 1 and 1.5.
r 0 (no wind)
r 0.25
r 0.5
r 1
r 1.5