457.562 Special Issue on River Mechanics (Sediment Transport)

(1)

Seoul National University

457.562 Special Issue on River Mechanics (Sediment Transport)

.02 Review of Fluid Mechanics

(2)

§ Shear stress between adjacent layers in a simple parallel flow is det ermined by the viscosity and the velocity gradient

§ If laminar flow is disturbed by an obstacle or roughness, then disturb ances must be damped by the molecular viscosity (stable)

§ However, in turbulent flow, inertia overcomes the viscous force and disturbance grows and becomes random motion.

§ Therefore, in turbulent motion rapid mixing of fluid particles during fl ow cause significant diffusion.

§ What is the actual meaning of viscosity? Is there any other name of viscosity?

1. Flow velocity distribution : law of the wall

τ = µ ^dv dy

(3)

§ Consider a steady, turbulent, uniform, open-channel flow having – H : a mean depth

– U : a mean flow depth

– B : a mean width (much greater than depth) – S : a mean slope

– k_s : effective height

– In a wide channel (B/H >>1)

1. Flow velocity distribution : law of the wall

(4)

§ The roughness height (or effective height) will be proportional to a m ean size or diameter D.

§ The bed shear stress

§ The above equation is the one-dimensional momentum conservatio n equation

§ Using the boundary shear stress, the shear velocity can be defined as

§ The sear velocity (the boundary shear stress) – provides a direct measure of the flow intensity – able to entrain and transport sediment particles.

1. Flow velocity distribution : law of the wall

u_* = τ_b ^/ ρ

τ_b = ρgHS (ρ = water density, g=gravity)

(5)

§ The above figure shows typical variation of total shear stress in a tur bulent pipe flow.

§ Based on the linear relationship of shear profile (in the previous clas s), and the Prandtl relationship,

τ = τ_b ¹− z H

⎛⎝⎜ ⎞

⎠⎟ = ρ^l² ^du dz

⎛⎝⎜ ⎞

⎠⎟

2

(6)

§ Near wall, l=kz, and mixing length in a pipe flow

§ Then the previous equation can be written

1. Flow velocity distribution : law of the wall

l = kz 1− z H

⎛⎝⎜ ⎞

⎠⎟

1/2

τ_b

ρκ ²^z² ⁼

du dz

⎛⎝⎜ ⎞

⎠⎟

2

du

dz = τ_b / ρ κ^z ⁼

u_* κ^z u

u_* = 1 κ ^ln

z z₀

⎛

⎝⎜

⎞

⎠⎟

u = time -averaged flow velocity at a distance z above the bed

z₀ = bed roughness length (distance above the bed where u(z₀) = 0) κ = is von Karman's constant ~0.41.

Only in a relatively thin layer (z/H=0.2) (1)

(7)

§ Through intensive experiment works,

§ When the effective roughness height becomes by (1) and (2)

§ Very near the smooth wall (viscous sublayer), velocity is linear with depth

§ Then two equations above need to match

1. Flow velocity distribution : law of the wall

u

u_* = 1 κ ^ln

u_*z

ν ⁺^5.5

u

u_* = u_*δ_ν ν ⁼

1 κ ^ln

u_*δ_ν

ν ⁺ ^5.5 ^{(at z} ⁼ δ_ν⁾ u_*δ_ν

ν ⁼^11.6 δ_ν =11.6 ν u u

u_* = u_*z

ν ^{(until z} ⁼δ_ν⁾ z_b =ν ^{/ 9u}_*

(2)

Hydraulic smooth condition of flow

(8)

§ Velocity distribution near a smooth wall

(9)

§ But, most boundaries in river flow are rough. Therefore we need to c onsider roughness height.

§ If k_s/δ_ν>1, then viscous sublayer will not exist.

§ The corresponding logarithmic velocity profile is given by

§ It follows that

§ Then what happens in the intermediate condition?

u

u_* = 1 κ ^ln

z k_s

⎛

⎝⎜

⎞

⎠⎟ +8.5 = 1

κ ^{ln 30} z k_s

⎛

⎝⎜

⎞

⎠⎟

z₀ = k_s / 30 Hydraulically rough flow

(10)

1. Flow velocity distribution : law of the wall

§ Hydraulically rough flow

§ Hydraulically smooth flow

§ Where “ the Roghness Reynolds number”

§ Yalin (1992) proposed a way to represent both ranges and intermedi ate ranges as

k_s

δ_ν 1, Re_* 11.6 k_s

δ_ν 1, Re_* 11.6

Re

_*

= u

_*

k

_s

ν , (R

^*C

= 11.6)

u

_*

= 1 κ ln

z k

_s

⎛

⎝⎜

⎞

⎠⎟ + B

_s

B

_s

= 8.5 + ⎡⎣ 2.5 ln Re ( )* − 3 ⎤⎦ e−0.121 ln Re[ *]2.42

(11)

§ Alternative way of writing,

u

_*

= 1

κ ln A

^s

z k

_s

⎛

⎝⎜

⎞

⎠⎟

A

_s

= e

^κ^B^s

(12)

2. Velocity-Defect and Log-Wakes laws

§ Equation (1) requires some knowledge of the bed roughness charact eristics.

§ An alternative formulation can be obtained if the flow depth H is intro duced as the relevant length scale.

§ Assuming that the maximum flow velocity u_max takes places at the w ater surface z=H, then equation (1) can be manipulated to obtain the velocity-defect law (outer low of the wall).

§ But, there were so many arguments and defects or log law working only in z/H<0.2. Nezu and Nakagawa added a wake function.

u_max −u

u_* = − 1 κ ^ln

z H

u

u_* = 1

κ ln u_*z

⎛ ν

⎝⎜ ⎞

⎠⎟ + 5.5+w z H

⎛⎝⎜ ⎞

⎠⎟

(3)

(13)

2. Velocity-Defect and Log-Wakes laws

§ W(z/H) is the wake function first proposed by Coles (1956) for turbul ent boundary-layer flows,

§ W₀ is the Coles wake parameter, expressing the strength of the wak e function.

§ Equation (3) can also be written in log-wake form

§ Coles wake parameter varies depending on the condition and incras es with increasing sediment concentration raging from 0.191 to 0.86 1.

w z H

⎛⎝⎜ ⎞

⎠⎟ = 2W₀

κ sin² π

2 z H

⎛⎝⎜ ⎞

⎠⎟

u_max −u

u_* = − 1

κ ln z H

⎛⎝⎜ ⎞

⎠⎟ + 2W₀

κ cos² πz 2H

⎛⎝⎜ ⎞

⎠⎟

(14)

2. Relations for channel resistance

§ Most river flows are indeed hydraulically rough.

§ The previous log law is used to obtain an appropriate expression for depth-averaged velocity U.

§ This relation is known as Keulegan’s resistance law for rough flow.

U = 1

H u dz

0 H

∫

U

u_* = 1 H

1 κ ^ln

z k_s

⎛

⎝⎜

⎞

⎠⎟ +8.5

⎡

⎣⎢ ⎤

⎦⎥dz

k_s H

∫

U u_* = 1

κ ^ln H k_s

⎛

⎝⎜

⎞

⎠⎟ +6 = 1

κ ^{ln 11} H k_s

⎛

⎝⎜

⎞

⎠⎟

(Avoiding z=0 : singular point)

(15)

2. Relations for channel resistance

§ Manning-Strickler form.

§ Log - > power : was first presented by Keulegan.

u

u_* = 1

κ ln 30 z k_s

⎛

⎝⎜

⎞

⎠⎟ ≅ 9.34 z k_s

⎛

⎝⎜

⎞

⎠⎟

1/6

U

u_* = 1

κ ln 11H k_s

⎛

⎝⎜

⎞

⎠⎟ ≅ 8.1 H k_s

⎛

⎝⎜

⎞

⎠⎟

1/6

(16)

2. Relations for channel resistance

§ Bed shear stress can be found as

§ In two-dimensional case τ_b = ρu_*² = ρC_fU²

C_f = 1

κ ln 11H k_s

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢ ⎤

⎦⎥

−2

or 8.1 H k_s

⎛

⎝⎜

⎞

⎠⎟

⎡ 1/6

⎣⎢

⎢

⎤

⎦⎥

⎥

−2

τ_bs,τ_bn

( )

⁼ ^ρ^C^f _⎡⎣^U² ⁺^V²_⎤⎦^1/2

(

^U^,V

)