Faculty of Coastal Engineering Faculty of Coastal Engineering
BED, BANK & SHORE BED, BANK & SHORE PROTECTION PROTECTION
Lecturer: PhamThu Lecturer:
Huong PhamThu Huong
Chapter 2 Chapter 2 Loads Flow -- Loads Flow
(3 class hours)
Content Content
2.1 Introduction
2.2 Turbulence
2.3 Wall flow
2.4 Free flow
2.5 Combination of wall flow and free flow
2.6 Load reduction
2.1 Introduction
forces
Structure
regular
Hour
tides
Flow
Fluctuation
Seconds
short waves
seconds
turbulence
Velocity field in various situations
averaged velocity values (ū = Q/A) Chezy's law for uniform flow: ū = C√(Ri)
Reynolds dye experiment Reynolds dye experiment
Reynolds number Reynolds number
vs - mean fluid velocity, L - characteristic length (h: water depth) μ - (absolute) dynamic fluid viscosity ν - kinematic fluid viscosity: ν = μ / ρ = 10-6 m2/s (water) ρ - fluid density
• Laminar flow occurs at low Reynolds numbers (Re<1000)
• Turbulent flow occurs at high Reynolds numbers (Re>2000)
2.2 Turbulence
Turbulence motion: velocity and pressure show irregular fluctuations
Turbulence variations:
′
′
′
′
u = u + u v = v + v w = w + w p = p + p
Turbulence variations:
′
′
′
′
u = u + u v = v + v w = w + w p = p + p
Turbulence can then be expressed in various ways, such as:
2
2
2
′
′
2
2
2
′
′
k
u
v
′ w
,
,
,
=
+
+
=
=
=
r u
r v
r w
(
)
1 2
u u
v u
′ w u
total kinetic energy in a turbulent flow
fluctuation intensities of u, v and w, relatively
Velocity Velocity
m * a = F
u
w
+
+
= −
+
u x
p x
u ∂ t ∂
∂ ∂
u ∂ z ∂
∂ ∂
2 u ∂ μ 2 z ∂
⎞ ⎟ ⎠
⎛ ρ ⎜ ⎝ inertia
press.
visc.
Reynolds stresses:
m * a = F
2
2
∂ u w ρ μ + + = − + − + u x u z p x u 2 z u ∂ t ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ′ u ∂ x ∂ ′ ′ u w z ∂ ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎛ ρ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠
inertia press. visc. Reynolds stresses
− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −
mean values turb. fluctuations
Exchange of momentum due to turbulence
2
Flow resistance:
u fc τ ρ=
In laminar flow the resistance, expressed as a shear stress, is proportional with the flow velocity.
In turbulent flow, the quadratic terms in equation become dominant and the relation between τand u becomes quadratic
Resistance in laminar and turbulent flow
2.3 Uniform wall flow 2.3 Uniform wall flow
Wall flow Wall flow
2
(
)
g h I
c
u
u
g h I
= τ ρ
=
ρ
=
=
ρ
⇒ =
2 u ρ *
b
f
' ' u w b b
1 c
f
u
g C /
=
u *
u* is the shear “velocity”
u C R I
C
Chezy:
with:
=
=
g c
f
c
2 / 3
1/ 6
Manning:
with:
u
R
I
=
n R =
f g
1 n
Nikuradse-Colebrook roughness:
C =
k (
is equivalent roughness )
r
R 12 ln k
R 12 18 log ≈ k
g κ
r
r
Example Example
20 m3/s of water flows in a 10 m wide channel with vertical banks, s, (cid:190)(cid:190) 20 m3/s of water flows in a 10 m wide channel with vertical bank a bed slope of 1/1000 and a roughness of 0.2 m. What is the a bed slope of 1/1000 and a roughness of 0.2 m. What is the Chezy--value, the relative turbulence depth, the velocity, the Chezy value, the relative turbulence depth, the velocity, the intensity and the relative turbulent shear stress? intensity and the relative turbulent shear stress?
Assume h
R =bh/(b+2h)
C=18log(12R/kr)
u= C√RI
Q* = bhu
Q* = Q
Stop
non uniform flow non uniform flow
The growth of a boundary layer when an infinitely thin plate is placed in a flow with u = u0.
Influence of pressure gradient on velocity Influence of pressure gradient on velocity profile profile
Uniform
accelerated
Decelerated
Turbulence in windtunnel Turbulence in
contraction windtunnel contraction
The total amount of turbulent kinetic energy, k, remains approximately constant.
Due to the increased velocity in the contraction, the relative turbulence, r, using the local mean velocity decreases.
2.4 Free flow 2.4 Free flow
50 100
Z
Flow, velocities and turbulence in mixing layer
Flow and velocities in jets Flow and velocities in jets
2
0.693
−
z b
⎛ ⎜ ⎝
⎞ ⎟ ⎠
3.5
⎛ ⎜ ⎜ ⎝
⎞ ⎟ ⎟ ⎠
Plane jets : u =
b =
0.1
x u = u e
m
m
u 0 x B
2
0.693
−
R b
⎛ ⎜ ⎝
⎞ ⎟ ⎠
⎛ ⎜ ⎜ ⎝
⎞ ⎟ ⎟ ⎠
Circular jets : u =
b =
x u = u e
0.1
m
m
u 6.3 0 x D
Turbulent fluctuations in circular jet Turbulent fluctuations in circular jet
instability of an axisymmetric instability of an
jet axisymmetric jet
effect of strong pressure gradients effect of strong pressure gradients
2.5 Combination of wall flow and Free flow 2.5 Combination of wall flow and Free flow
Flow separation around blunt and round body
Vertical constriction and expansion (sill) Vertical constriction and expansion (sill)
Vertical expansion Vertical expansion facing step) (backward--facing step) (backward
Horizontal expansion Horizontal expansion
Horizontal constriction and expansion Horizontal constriction and expansion (groyne)) (groyne
