457.309 Hydraulics and Laboratory .06 Hydraulic Scale Models Prepared by Jin Hwan Hwang (2019.03.12)

Seoul National University

457.309 Hydraulics and Laboratory .06 Hydraulic Scale Models

Prepared by Jin Hwan Hwang (2019.03.12)

Seoul National University

Hydraulic Scale Models

§ Model vs. Real Natural Flow

–

Natural flow is much more complex than modeled flow.

–

Natural flow is much larger than modeled flow.

§ How we can mimic real flow with model?

–

Following dimensional numbers…

Seoul National University

Tidal control gate

§ Maeslant Barrier

3

Seoul National University

Why do quantities between model and prototype disagree?

§ Measurement effects:

– due to non-identical measurement techniques used for data sam pling in the model and prototype (intruding versus non-intruding measurement system etc.).

§ Model effects:

– Due to the incorrect reproduction of prototype features such as g eometry (2D modelling, reflection from boundaries), flow or wave generation techniques or fluid properties

§ Scale effects:

– Due to the inability to keep each relevant force ratio consta nt between the he scale model and its real-world prototype .

4

Seoul National University

Example of scale effects

5

Seoul National University

Types of Similarity

§ If model fulfill the similarity condition, then we can use model to study natural flows.

§ Three types of similarity

– Geometric

– Kinematic – Dynamic

Seoul National University

Geometric similarity

§ Different scale, generally smaller

– Length

– Area

– Volume

Seoul National University

Kinematic similarity

§ Similarity of motion

–

Stream line, boundary layer, etc

–

Kinematic similarity must ensure geometric similarity

–

Where a is the acceleration

1 2 1

2 1

2

.

- -

-

=

=

=

=

=

=

=

=

=

=

r r r

r p

m r

r r p

m r

r r r

r p

r m

p m r

L V T

a V a a

V T L

T T

V L V

Q A Q Q

Also

const V

V V

Seoul National University

Dynamic similarity

§ Dynamic -> Force similarity !

§ Three main forces

–

External force: Gravity force

–

Internal forces: Viscous and surface tension)

–

Resultant force : drag force, pressure force on hydraulic structures

§ Those main forces ratio must be equivalent in both model and

prototype (real)

Seoul National University

Dynamic similarity

§ Assuming that F

_I

, F

_g

, F

_n

, F

_s

, F

_p

, and F

_Ie

, are the inertia, gravity, viscous, surface tension, pressure, and elastic force respectively

–

Then

–

Regardless of the specific type of the force, its dimensions are

equivalent to the dimensions of mass (M) multiplies by acceleration (a).

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) F F const .

F F F

F F

F F

F F

e p e m p

m p

m g p

g m I p

I m

r

= = = = = =

s s n

n

( )(

r ³r r r ¹

)

r ²r r

(

r r ¹

)

r ²r r² r

r

r

M a L V T L V L T L V

F = = r

^-

= r

^-

= r

Seoul National University

Dynamic similarity

– Work – Power – Pressure

– Specific gravity

– It is clean that the ratios of all parameters in two systems, we can express with 3 main parameter ratios of geometric (length),

kinematic(velocity), and dynamic(density).

– While length and density are given in the laboratory, the velocity is not determined well… We need to determine the dominating force.

2 3

r r r r

r

r

F L L V

W = = r

3 2 2

2

r r r r

r r r r

r

r

F V L V v L V

P = = r × = r

2 2

2 2 2

r r r

r r r r

r

r

F L L V L V

p =

^-

= r ×

^-

= r

2 1 3

r r r r

r

r

= F L

^-

= r L

^-

V g

p r

r

m

r = r

Seoul National University

Dominating forces

§ In hydraulic problems, there are a number of forces exist. Howeve r, only one or two forces dominate.

§ To reduce the cost and simplify the solution, we generally consider

one or two forces.

Seoul National University

Gravity force

§ In open channels, natural rivers, hydraulic structures, wave actio ns, etc, gravity force is dominant.

§ Or

§ Where F

_I

= r L

²

V

^2;

and F

_g

= g L

³

§ Which can be written as

g p I g m

I

F F F

F ÷ ÷

ø ö ç ç

è

= æ

÷ ÷ ø ö ç ç

è æ

= 1

÷ ÷ ø ö ç ç

è æ

g r I

F F

3

1

2

2

÷÷ =

ø çç ö

è æ

L

r

V L g r

( )

n _r r

gL F

V

² ₂

1 =

÷÷ = ø çç ö

è

æ

Seoul National University

Gravity force

§ The above equation indicates that when gravity force dominates , the Froude number in the model must be exactly equal to that i n the prototype.

§ The velocity ratio for gravity problems

§ Gravity does not change in model and real world, so

( )

n _r r

gL F

V

² ₂

1 =

÷÷ = ø çç ö

è æ

( ) ÷ ÷ = 1

ø ö ç ç

è

= æ

r n r

gL F V

r r

r

g L

V =

r

r

L

V =

Seoul National University

Example: A 1:10 scale model is constructed for a hydraulic structure. Determine the ratios of force and pressure.

If the force and pressure on the model are equal to 1.90kN and 500N/m

²

, respectively, what are the corresponding values in the prototype? Assume the same fluid is used in both the model and prototype.

§ Solution:

Using force ration equation where

10

= 1 Lr

2 2

r r r

r

L V

F = r

2 /

,

1

1

_{r r}

r

= V = L

r

Seoul National University

Example: A 1:10 scale model is constructed for a hydraulic structure. Determine the ratios of force and pressure.

Then F

_p

=1.9X1000=1900kN.

2 2

r r r

r

L V

F = r

p p

m r

r r r

r r r r

r r

r r

F F

F p

V L

V A L

p F L F

9 . 1 1000

1 10

1

1000 1

3 2

3 2 3

=

=

=

=

=

=

=

=

-

r

r r

_r

= 1 , V

_r

= L

¹_r^/2

/ 2

5

500 10

1 m kN p

p p

p p

p

p p

m r

=

=

=

=

Seoul National University

Viscous Force

§ Viscous force mainly appear in the pipe flow, or partially submerged object.

– Viscous force dominant means that flow is laminar

– In the opposite way, if viscous force is ignored, what kind of flow can occur?

Turbulent flow Inviscid flow

§ Following Newton’s law of viscosity

t is shear stress due to viscosity of µ which is dynamic viscosity

dy

µ dv

t =

Seoul National University

Viscous Force

§ Following Newton’s law of viscosity

t is shear stress due to viscosity of µ which is dynamic viscosity

§ Rewrite in dimensional form, then

For dynamic similarity, the ratio of two main forces of inertia and viscosity must be equivalent.

dy µ dv t =

vL F

L v L

F

µ µ

n n

=

\

2

=

Seoul National University

§ For dynamic similarity, the ratio of two main forces of inertia and viscosity must be equivalent.

§ Let’s think in the different way,

It means that the Reynolds number is the force balance of inertia and viscous

Viscous Force

1

.

÷÷ = ø çç ö è æ

÷÷ = ø çç ö è

= æ

÷÷ ø çç ö è

\ æ

r I

p I m

I

F or F

const F

F F

F

n

n n

I

LV LV R

n

VL V L F

F = = = =

µ µ

r µ

r

n

2 2

Seoul National University

§ Then we can conclude that for dynamically similar system with a viscous force dominating the flow, we have

§ Reynolds number ratio of model and prototype (real) must be 1.

§ Let’s say we use the same fluids in both systems, then viscosity must be equal.

Viscous Force

( ) ÷÷ = 1

ø çç ö

è

= æ

r n r

R LV

µ r

1 1

1 - -

-

=

=

_{r r r r r}

r

L L

V µ r n

1

1

1

^-

-

= ×

=

_{r r r}

r

L L

V n

Seoul National University

§ Let’s think drag forces on floating body like ships. Two component are important (for design or experiment)

1.

Wave resistance (gravitational force)

2.

Viscous resistance (viscosity)

3.

(the importance of inertia need not to be discussed)

§ Using dimensional analysis, the drag force F, the fluid velocity V, a characteristic length of the body, L, density r, viscosity, m, the gravitational acceleration g,

Gravity and viscous forces

( F

_n

R

_n

)

V L

F = r

^{2 2}

j ,

Seoul National University

§ For the dynamic similarity, both F

_n

and R

_n

should be the same value in the two systems.

In viscous relationship, we drove In gravity relationship,

Both system has same gravitational acceleration, then

§ So, if you need to reduce size of model then use different material for experiment from in the real world

Gravity and viscous forces

( ) ( ) ( )

( ) ( ) ( ) 1

1

=

=

=

=

n r n p

n m

n r n p

n m

R or

R R

F or

F F

1 1

1 - -

-

=

=

_{r r r r r}

r

L L

V µ r n

r r

r

g L

V =

2 / 3 1

2 /

1_{r r}

L

_r

, v

_r

L

_r

L = n

^-

=

Seoul National University

§ If we cannot maintain full geometric similarity for economic, practical or physical, we should use distorted model.

☞ Roughness height is impossible to change…

§ For example, 10km river with 1km width and 4m depth.

§ But, hydraulic laboratory has only dimension of 100m ´30m.

Then we need

Distorted models

) 4

100 ( 1

) 10

100 ( 1 100

1

H for H cm

L H

B for B m

L B

length for

L

p r m

p m r

r

=

=

=

=

\

=

Seoul National University

§ When H=4cm, surface tension may be important. But, in real situation, surface tension is not crucial.

§ So, in the practical perspectives, we consider two different scales for vertical and horizontal.

Distorted models

) 4

100 ( 1

) 10

100 ( 1 100

1

H for

H cm L H

B for B m

L B

length for

L

p r m

p m r

r

=

=

=

=

\

=

Seoul National University

Distorted model

§ “… are used extensively in rivers and harbors as flow depths are quite small as compared to other areal

dimensions.

§ When Z

_r

is taken as 1:100, while L

_r

is taken as 1:200 or

even 1:500.

Seoul National University

Distorted model

§ In distorted models, we will only discuss gravity force as a dominating force.

§ Most distorted models are built for rives, harbors, waves and other systems where flow mainly controlled by the gravity force.

§ The above equation is similar to before, we use here Z instead of L. since fluid velocities are created by vertical heads.

2 / 2 1 / 1

r r

r

g Z

V =

r r

r

g L

V =

Seoul National University

Distorted model

§ The gravity acceleration can be assumed as constant.

Then simply

§ Be careful whether we consider ratios for horizontal or vertical. For example, the ratio for a horizontal area A

_rh

is

§ The ratio for the vertical area A

_rv

is

2 / 1 r

r

Z

V =

2 r

r

L

A

h

=

r r

r

L Z

A

v

=

Seoul National University

Distorted model

§ In distorted models, we will only discuss gravity force as a dominating force.

§ Most distorted models are built for rives, harbors, waves and other systems where flow mainly

controlled by the gravity force.

§ The above equation is similar to before, we use here Z instead of L. since fluid velocities are created by vertical heads.

2 / 2 1 / 1

r r

r

g Z

V =

r r

r

g L

V =

Seoul National University

Distorted model

§ Discharge

§ Time

§ Force

§ Work

2 / 2 1

2 / 3 2

/ 1

r r r

r r

r r r

r r r

r r

Z L V

A Q

Z L Z

Z L V

A Q

h v

v h

=

×

=

=

×

=

×

=

2 / 1

2 / 1

r r

r r

r r r

r r

V Z T Z

Z V L

T L

v h

=

=

=

=

^-

( )

r r r r

r r r

r r r

r r r r

r r r r

r r r

r r r

Z L V

A V

Q F

Z L Z

Z L V

A V

Q F

h v

v

v h

h

2 2

2 2

r r

r

r r

r r

=

=

=

=

=

=

=

2 2 2

r r r r

r r r r

r r r

Z L L

Z L W

L F W

h

h h

r

r × =

=

×

=