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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:. - PowerPoint PPT Presentation
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1
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.
ijj
i2
ij
ij
i gxx
u
x
p1
x
uu
t
u
0x
u
i
i
2
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
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
ij
i
jji
ji
i gx
u
xuu
xx
p
t
u
A B C D E
3
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:
0z
w
y
v
x
u
z2
2
2
2
2
2
y2
2
2
2
2
2
x2
2
2
2
2
2
gz
w
y
w
x
w
z
p1
z
ww
y
wv
x
wu
t
w
gz
v
y
v
x
v
y
p1
z
vw
y
vv
x
vu
t
v
gz
u
y
u
x
u
x
p1
z
uw
y
uv
x
uu
t
u
4
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThe 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.
x
y
U0z
w
y
v
x
u
z2
2
2
2
2
2
y2
2
2
2
2
2
x2
2
2
2
2
2
gz
w
y
w
x
w
z
p1
z
ww
y
wv
x
wu
t
w
gz
v
y
v
x
v
y
p1
z
vw
y
vv
x
vu
t
v
gz
u
y
u
x
u
x
p1
z
uw
y
uv
x
uu
t
u
Thus for any constant rectilinear flow, all that needs to be satisfied is the hydrostatic pressure distribution (even though there is flow):
z
y
x
gz
p10
gy
p10
gx
p10
or ii
gx
p
5
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
y
x
u
moving with velocity U
fixed
fluid
xy2112
H
6
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThat is, the components of the viscous stress tensor are
y
x
u
moving with velocity U
fixed
fluid
000
00dy
du
0dy
du0
z
w2
y
w
z
v
x
w
z
u
y
w
z
v
y
v2
x
v
y
u
x
w
z
u
x
v
y
u
x
u2
vij
Here we abbreviate dy
duxy
H
7
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThus u = u(y) only, and v = w = 0. This result automatically satisfies continuity:
0z
w
y
v
x
u
Momentum balance in the x, y and z directions (z is upward vertical)
z2
2
2
2
2
2
y2
2
2
2
2
2
x2
2
2
2
2
2
gz
w
y
w
x
w
z
p1
z
ww
y
wv
x
wu
t
w
gz
v
y
v
x
v
y
p1
z
vw
y
vv
x
vu
t
v
gz
u
y
u
x
u
x
p1
z
uw
y
uv
x
uu
t
u
g
8
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWMomentum balance in the z direction (out of the page):
y
x
u
moving with velocity U
fixed
fluid
gdz
dpg
dz
dp10
That is, the pressure distribution is hydrostatic. Recall that the general relation for a pressure distribution ph obeying the hydrostatic relation is:
ii
gx
p
H
9
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWMomentum balance in the x (streamwise) direction:
y
x
u
moving with velocity U
fixed
fluid
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)
Uu,0uHy0y
dy
duwhere0
dy
dor
dy
ud0
2
2
H
10
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWIntegrate once:
y
x
u
moving with velocity U
fixed
fluid
Thus the shear stress must be constant on the domain.
Uu,0uHy0y
dy
duwhereCorC
dy
du11
H
Integrate again:
Apply the boundary conditions to obtainC2 = 0, C1 = U/H and thus
21 CyCu
H
U
H
U,
H
yUu
11
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.
Hx
z
u
12
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 << 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
BThe vector of gravitational acceleration is (gx, gy, gz) = (gsin, 0, -gcos)
Hx
z
u
g
gcos
gsin
13
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Hx
z
u
Continuity is satisfied if u = u(z) and v = w = 0.
0z
w
y
v
x
u
The equations of conservation of streamwise and upward normal momentum reduce to:
cosgz
w
y
w
x
w
z
p1
z
ww
y
wv
x
wu
t
w
singz
u
y
u
x
u
x
p1
z
uw
y
uv
x
uu
t
u
2
2
2
2
2
2
2
2
2
2
2
2
14
Hx
z
u
The equations thus reduce to:
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
cosgdz
dp10
singdz
ud0
2
2
Since
dz
du
x
w
z
u3113
The first equation can thus be rewritten as
singdz
d0
where is an abbreviation for 13 = 31.
15
Hx
z
u
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
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
cosgdz
dp,sing
dz
ud
dz
d2
2
are
0p
0
0u
Hz
Hz
0z
viscous fluid sticks to immobile bed
no applied shear stress as free surface
gage pressure at free surface = 0 (surface pressure = atmospheric)
16
Now the condition
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
cosgdz
dp
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
0pHz
yields a pressure distribution that varys linearly in z:
H
z,)1(cosgHp
Hx
z
u
p
17
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The equation
singdz
d
subject to
0Hz
similarly yields a linear distribution for shear stress in the z direction:
H
z,)1(singH
Hx
z
u
Note that the bed shear stress b at z = 0 is given as
singHb
18
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
Thus
H
z1singH
dz
du
subject to
0u0z
Integrates to give the following parabolic profile for u in z:
H
z,
2
1sinH
gu 22
Hx
z
u
19
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOW
The maximum velocity Us is reached at the free surface, where z = H and = 1);
Thus
H
z,
2
12
U
u 2
s
Hx
z
u
sinHg
2
1U 2
s
Depth-averaged flow velocity U is given as 1
0
H
0ududz
H
1U
Thus
2
3
U
U,sinH
g
3
1U s2
20
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 D’arcy-Weisbach friction coefficient. Between the above relation and the relations below
Hx
z
u
2b
f UC
it can be shown thatHere Re denotes the dimensionless Reynolds No. of the flow, which scales the ratio of inertial forces to viscous forces.
,sinHg
3
1U 2
singHb
UH,
3Cf Re
Re
21
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
wHz
Hx
z
w
u
The corresponding solution to the problem is
,)r1(singH H
z,
2
1)r1(sinH
gu 22
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:
Hsingr w
22
LAMINAR PLANE COUETTE AND OPEN CHANNEL FLOWThe solution for velocity with the case of wind can be rewritten as
where und is a dimensionless velocity equal to 2u/(gsinH2).
,H
z,
2
1)r1(2u 2
nd
A plot is given below of und versus for the cases r = 0. 0.25, 0.5, 1 and 1.5.
Hsingr w
Velocity Distribution with Wind
0
0.2
0.4
0.6
0.8
1
-2 -1.5 -1 -0.5 0 0.5 1
und
r = 0(no wind)r = 0.25r = 0.5r = 1r = 1.5