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Hydromechanics: Course Summary

Hydromechanics VVR090

Material Included;

• French: Chapters 1 to 9 and 14 + Sample problems

• Vennard & Street: Chapters 8 + 13, and 11 (part of it)

• Roberson & Crowe: Chapter 11

• Collection of sample problems in open channel flow

Exam:

28th of May 1400 – 1900 in MA10G

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Fundamental Equations

Conservation of mass:

Q uA=

Conservation of momentum:

2 1( )F Q u u= ρ −∑

Conservation of energy:

2

1 2

2

L

uH z yg

H H h

= + +

= +

Laboratory Experiments

Often difficult to solve fluid flow problems by analytical or numerical methods. Also, data are required for validation.

The need for experiments

Difficult to do experiment at the true size (prototype), so they are typically carried out at another scale (model).

Develop rules for design of experiments and interpretation of measurement results.

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Similitude and Dimenisional Analysis

Similitude: how to carry out model tests and how to transfer model results to prototype (laws of similarity)

Dimensional analysis: how to describe physical relationships in an efficient, general way so that the extent of necessary experiments is minimized (Buckingham’s P-theorem)

Basic Types of Similitude

• geometric

• kinematic

• dynamic

All of these must be obtained for complete similarity between model and prototype.

= = λp p

m m

d ld l

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Important Forces for the Flow Field

• pressure (FP)

• inertia (FI)

• gravity (FG)

• viscosity (FV)

• elasticity (FE)

• surface tension (FT)

Dimensionless Numbers

• Reynolds

• Froude

• Cauchy (Mach)

• Weber

• Euler

2 22

2

2

Re/

Fr

C/

W

E2

= =μ ρ ν

=

= = =ρ

ρ=

σρ

=Δ

Vl Vl

Vgl

V V ME c

lV

Vp

Dimensionless numbers same in prototype and model produces dynamic similarity.

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Dimensional Analysis

Dimensions (e.g., length, mass, time, temperature)

Units (e.g., m, kg, s, K)

Three independent dimensions of primary interest:

• length (L)

• mass (M)

• time (t)

Force: [ ] 2=MLFt

Metric system

Buckingham’s P-Theorem

Buckingham provided a systematic approach to dimensional analysis through his theorem expressed as:

1. If n variables are involved in the problem, then k equations of their exponents can be written

2. In most cases k is the number of independent dimensions (e.g., M, L, t)

3. The functional relationship may be expressed in terms of n-k distinct dimensionless groups

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Example of Dimensional Analysis

Drag force (D) on a ship. Assume that D is related to length (l), density (ρ), viscosity (m), speed (V), and acceleration due to gravity (g):

{ }, , , , , 0ρ μ =f D l V g

Problem involves n = 6 variables and k = 3 fundamental dimensions Æ k - n = 6 – 3 = 3 dimensionless groups can be formed:

{ }1 2 3' , , 0Π Π Π =f

Many different ways to combine the variables into dimensionless groups – rational approach needed.

Method for Deriving Dimensionless Groups

1. Find the largest number of variables which do not form a dimensionless P-group

2. Determine the number of P-groups to be formed

3. Combine sequentially the variables in 1. with the remaining variables to form P-groups.

Present example: select ρ, V, and l and combine with remaining variables:

{ }{ }{ }

1 1

2 2

3 3

, , ,

, , ,

, , ,

Π = ρ

Π = μ ρ

Π = ρ

f D V l

f V l

f g V l

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First P-group:

1Π = ρa b c dD V l

Analyze dimensions:

( )0 0 02 3

: 0: 0 3

: 0 2

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= += − + += +

a b cdML M LM L t L

t L t

M a bL a b c dt a c

22

= −= −= −

b ac ad a

1 2 2

⎛ ⎞Π = ⎜ ⎟ρ⎝ ⎠

aD

l VResult:

Fluid Flow About Immersed Objects

Flow about an object may induce:

• drag forces

• lift forces

• vortex motion

Asymmetric flow field generates a net force

Drag forces arise from pressure differences over the body (due to its shape) and frictional forces along the surface (in the boundary layer)

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sin cos= θ + τ θ∫ ∫ oA A

D p dA dA

Drag force:

Pressure drag (Dp) Frictional drag (Df)

(form drag)

Pressure drag function of the body shape and flow separation

Frictional drag function of the boundary layer properties (surface roughness etc)

Results of Dimensional Analysis

{ }

2

3

12Re,M

D o

D

D C AV

C f

= ρ

=

Total drag force:

Total lift force:

{ }

2

4

12

Re,M

L o

L

L C AV

C f

= ρ

=

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Drag Coefficient for Various Bodies

2D 3D

Example of Drag Force Calculation

• parachute jumping

• sedimentation of particle

• popcorn popper

Basic equation for drag force:

212

= ρD oD C AV

CD obtained from empirical studies

A is the projected area on a plane perpendicular to the flow direction

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Vortex Shedding

Under certain conditions vortices are generated from the edges of a body in a flow.

Æ Von Karman’s vortex street

Vortex street behind a cylinder

Vortices at Aleutian IslandIf 6 < Re < 5000, regular vortex sheeding may occur at a frequency ndetermined by Strouhal’s number:

=o

ndSV (S = 0.21 over a wide range of Re)

Boundary Layer on a Flat Plate

Boundary layer: the zone in which the velocity profile is governed by frictional action

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Drag Coefficient for Smooth, Flat Plates

212f f oD C V A= ρ

A: surface area of plate

Open Channel Flow

Open channel: a conduit for flow which has a free surface

Free surface: interface between two fluids of different density

Characteristics of open channel flow:

• pressure constant along water surface

• gravity drives the motion

• pressure is approximately hydrostatic

• flow is turbulent and unaffected by surface tension

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Flow Classification

• steady – unsteady

• uniform – non-uniform

• varied flow (= non-uniform):

gradually varied – rapidly varied

Flow Classification

subcritical – supercritical flow

characterized by the Froude number

UFrgL

=

L taken to be the hydraulic depth D=A/T

Fr < 1 subcritical flowFr = 1 critical flowFr > 1 supercritical flow

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Definition of Channel and Flow Properties

Hydraulic radius (R): ratio of flow area to wetted perimeter

ARP

=

ADT

=

Hydraulic depth (D): ratio of flow area to top width

Energy Equation

2

2= + +

γp uH z

g

Total energy of a parcel of water traveling on a streamline (no friction):

velocity headpressure head

elevation head

+γpz hydraulic grade line

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Critical Flow

2 2

22 2= +α = +α

u QE y yg gA

Specific energy:

2

1

1

=

= =

ugD

uFrgD

Minimum specific energy yields:

Critical Flow

Rectangular channel of width b:

=

=

Qqbquy

1/32

2

2 223

⎛ ⎞= ⎜ ⎟⎝ ⎠

=

=

c

c c

c c

qyg

u yg

y E

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Step in Rectangular Channel

2 21 2

1 2

2 1

2 2u uy y zg g

E E z

+ = + + Δ

= −Δ

Bernoulli equation (between upstream and downstream points):

Water Surface Variation from the Energy Equation

Total energy:

2

2uH z yg

= + +

Differentiating with respect to distance:

( )2 / 2= + +

d u gdH dz dydx dx dx dx

21−

=−o fS Sdy

dx FrResulting equation:

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Momentum Equation

Momentum equation (rectangular channel):

( )2 21 2

2 12 2y y q u u

gγ γ γ

− = −

Hydraulic jump

Momentum equation for rectangular section:

( )2

2 22 1

1 2

1 1 12

q y yg y y⎛ ⎞

− = −⎜ ⎟⎝ ⎠

Solutions:

( )

( )

221

1

212

2

1 1 8 12

1 1 8 12

y Fry

y Fry

= + −

= + −

Energy loss:( )32 1

1 24y y

Ey y−

Δ =

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Uniform Flow

Uniform occurs when:

1. The depth, flow area, and velocity at every cross section is constant

2. The energy grade line, water surface, and channel bottom are all parallel:

f w oS S S= =

Sf = slope of energy grade line

Sw = slope of water surface

So = slope of channel bed

Uniform Flow Formula

Mannings equation for velocity:

2/31u R Sn

=

Uniform flow rate:

2 /31Q uA AR Sn

= =

Section factor:

Conveyance:

2 /3AR

2/31K ARn

=

(increases with depth)

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Computation of Uniform Flow

1. Channel cross section and shape, water depth, and slope known => Q or u can be calculated directly

2. Channel cross section and shape, water velcoity or flow, and slope known => water depth may be calculated through some iterative procedure

Roughness known and constant.

0.012

0.014

0.016

Manning’s Roughness n

0.018

0.018

0.020

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Gradually Varied Flow

Depth of flow varies with longitudinal distance.

Occurs upstream and downstream control sections.

Governing equation:21

−=

−o fS Sdy

dx Fr

Classification of Gradually Varied Flow Profiles

Prevailing conditions:

If y < yN, then Sf > So

If y > yN, then Sf < So

If Fr > 1, then y < yc

If Fr < 1, then y > yc

If Sf = So, then y = yN

Water surface profiles may be classified with respect to:• the channel slope • the relationship between y, yN, and yc

Profile categories:M (mild) 0 < So < ScS (steep) So > Sc > 0C (critical) So = ScA (adverse) So < 0

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Gradually Varied Flow Profile Classification II

Flow Transition

Subcritical to supercritical

Supercritical to subcritical

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Strategy for Analysis of Open Channel Flow

1. Start at control points

2. Proceed upstream or downstream depending on whether subcritical or supercritical flow occurs, respectively

Typical approach in the analysis:

Control points typically occur at physical barriers, for example, sluice gates, dams, weirs, drop structures, or changes in channel slope.

Uniform Channel

Prismatic channel with constant slope and resistance coefficient.

Apply energy equation over a small distance Dx:

2

2 o fd uy S Sdx g

⎛ ⎞+ = −⎜ ⎟

⎝ ⎠

Express the equation in difference form:

( )2

2 o fuy S S xg

⎛ ⎞Δ + = − Δ⎜ ⎟⎝ ⎠

2 2

4/3fn uSR

=

22

Dxi

Reach i

x

yi yi+1

( ) ( )( )

2 2

12 2 4 /3

1/ 2

/ 2 / 2

/i i

io i

y u g y u gx

S n u R+

+

+ − +Δ =

−

All quantities known at i. Assume yi+1 and compute Dxi (ui+1 given by the continuity equation).

ui

ui+1

Computation of Gradually Varied Flow

Trial-and-Error Approach

Well-suited for computations in non-prismatic channels.

Channel properties (e.g., resistance coefficient and shape) are a function of longitudinal distance.

Depth is obtained at specific x-locations.

Apply energy equation between two stations located Dxapart (z is the elevation of the water surface):

2

2 21 2

1 2

2

2 2

f

f

uz S xg

u uz z S xg g

⎛ ⎞Δ + = − Δ⎜ ⎟⎝ ⎠

+ = + + Δ

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Equation is solved by trial-and-error (from 2 to 1):

1. Assume y1 Æ u1 (continuity equation)

2. Compute Sf

3. Compute y1 from governing equation. If this value agrees with the assumed y1, the solution has been found. Otherwise continue calculations.

Estimate of frictional losses:

( )1 212f f fS S S= +

Examples of Gradually Varied Flow

Flow in channel between two reservoirs (lakes):

1. Steep slope, low downstream water level

2. Steep slope, high downstream water level

3. Mild slope, long channel

4. Mild slope, short channel

5. Sluice gate located in the channel

Study flow situation that develops + calculation procedure

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Spatially Varied Flow

Flow varies with longitudinal distance.

Examples: side-channel spillways, side weirs, channels with permeable boundaries, gutters for conveying storm water runoff, and drop structures in the bottom of channels.

Two types of flow:

• discharge increases with distance

• discharge decreases with distance

Different principles govern => different analysis approach

Increasing discharge: use momentum equation

(hard to quantify energy losses)

Decreasing discharge: use energy equation

2

22QH z ygA

= + +

2 21 1 ( ) ( ) 02 2o a a ay b y b Q x u xγ − γ = ρ −

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Weirs

Types of weirs (classified according to shape):

• rectangular

• V-notch

• trapezoidal

• parabolic

• special type (e.g., Cipoletti, Sutro)

Distinguish between:

• Broad-crested

• Sharp-crested

Discharge Formula for Rectangular Broad-Crested Weir

Apply Bernoulli equation between upstream section and the control section (critical depth occurs here).

h1

1/ 23/ 21

2 23 3D vQ C C g Th⎛ ⎞= ⎜ ⎟⎝ ⎠

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Discharge Formula for Sharp-Crested Weirs

h1

z

( )1/ 2 3/ 21

2 23 eQ C g bh=

( ) ( )1/ 2 5/ 21

8 2 tan / 215 eQ C g h= θ

Rectangular:

Triangular:

Parshall Flume

Critical flow occurs in the flumethroat followed by a hydraulic jumpdownstream.

BaQ AWH=General discharge formula:

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Venturimeters

Involves a constriction in the flow. The constriction produces an accelerated flow and a fall in the hydraulic grade line (pressure) directly related to the flow rate.

( )2 1 2

1 222 1

21 /

vC A p pQ g z zA A

⎛ ⎞= + − −⎜ ⎟γ γ⎝ ⎠−

Orifices

Used for many purposes in engineering, including measuring the flow rate.

A difference compared to nozzles is that the minimum flow section does not occur at the orifice but some distance downstream (in vena contracta).

Flow rate is given by:

( )1 2

1 2221

21 /

v c

c

C C A p pQ g z zC A A

⎛ ⎞= + − −⎜ ⎟γ γ⎝ ⎠−

Area in vena contracta is:2 cA C A=

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Submerged Orifice

Discharge from one large reservoir to another:

2 2c vQ C C A gh CA gh= =

Discharge to the atmosphere:

( )1 22c vQ C C A g h h= −

Sluice Gate

Special case of orifice flow: only contraction on the top of the jet.

Pressure in vena contracta is assumed to be hydrostatic.

( )( )1 22

2 1

21 /

v cC C AQ g y yy y

= −−