EAH 225Dec 28th

HYDRAULICS

Dr H Md AzamathullaLecturer,REDAC, USM

Information for the homework/assignments:Working with other students• You have to work on the homework problems yourself

first for a reasonable time• After you have tried yourself, you are encouraged to

work in groups on solving the problems (fellow students in discussion sections, lab partners)

– Benefits to the explainer: Learn to explain how to solve a problem, then you are sure you really understand it.

– Note of caution: Do not let others solve problems for youwithout having tried yourself: You need to learn how to solve problems, not just how to plug into equations to pass the exams• Carry the calculations out independently, explain inyour own words in the HW you hand in on paper.

Dimensional Analysis

• Both sides of equation must have the sameDimensions

-> Check whether dimensions on both sides ofan equation match! – this is one way ofDimensional Analysis

A technique for converting from one unit to another

Conversion Factors- In dimensional analysis, we makeconversion factors into fractions that wewill multiply by.- For example, one conversion factor is:1 inch = 2.54 cm- We can make (2) fractions out of this…1 inch OR 2.54 cm---------- ------------2.54 cm 1 inch

If the perimeter P of alluvial channel is related to the channel forming discharge Q by the equation

P= 2.667 (Q)0.5 where P is in feet and Q is in feet3/sec , modify the formula for its use in SI units.

P=2.66 (Q)0.5

Nature of Dimensional AnalysisExample: Drag on a Sphere

Drag depends on FOUR parameters:sphere size (D); speed (V); fluid density (ρ); fluid viscosity (μ)Difficult to know how to set up experiments to determine dependenciesDifficult to know how to present results (four graphs?)

Nature of Dimensional AnalysisExample: Drag on a Sphere

Only one dependent and one independent variableEasy to set up experiments to determine dependencyEasy to present results (one graph)

Buckingham Pi Theorem

• Step 1:List all the dimensional parameters involved

Let n be the number of parameters

Example: For drag on a sphere, F, V, D, ρ, μ, and n = 5

Buckingham Pi Theorem

• Step 2Select a set of fundamental (primary) dimensions

For example MLt, or FLt

Example: For drag on a sphere choose MLt

Buckingham Pi Theorem• Step 3

List the dimensions of all parameters in terms of primary dimensions

Let r be the number of primary dimensions

Example: For drag on a sphere r = 3

Buckingham Pi Theorem

• Step 4Select a set of r dimensional parameters that includes all the primary dimensions

Example: For drag on a sphere (m = r = 3) select ρ, V, D

Buckingham Pi Theorem• Step 5

Set up dimensional equations, combining the parameters selected in Step 4 with each of the other parameters in turn, to form dimensionless groups

There will be n – m equations

Example: For drag on a sphere

Buckingham Pi Theorem• Step 5 (Continued)

Example: For drag on a sphere

M a+1=0 ---- a=-1

t -b-2=0 ------ b=-2

L -3a+b+c+1=0

Substituting for a and b, +3-2+c+1=0

Therefore c=-2

Buckingham Pi Theorem• Step 6

Check to see that each group obtained is dimensionless?

Example: For drag on a sphere

Any remaining groups are produced in the same way, and thus П2 =µ/ρVD

Significant Dimensionless Groups in Fluid Mechanics

• Reynolds Number

Froude Number F =V/(gD)0.5

Significant Dimensionless Groups in Fluid Mechanics

• Euler Number

Cavitation Number

Flow Similarity and Model Studies

• Geometric Similarity– Model and prototype have same shape– Linear dimensions on model and prototype correspond

within constant scale factor• Kinematic Similarity

– Velocities at corresponding points on model and prototype differ only by a constant scale factor

• Dynamic Similarity– Forces on model and prototype differ only by a

constant scale factor

Flow Similarity and Model Studies

• Example: Drag on a Sphere

Flow Similarity and Model Studies

• Example: Drag on a Sphere

For dynamic similarity …

… then …

Flow Similarity and Model Studies

• Incomplete Similarity

Sometimes (e.g., in aerodynamics) complete similarity cannot be obtained, but phenomena may still be successfully modelled

Flow Similarity and Model Studies

• Scaling with Multiple Dependent Parameters

Example: Centrifugal Pump

Pump Head

Pump Power

Flow Similarity and Model Studies

• Scaling with Multiple Dependent Parameters

Example: Centrifugal Pump

Head Coefficient

Power Coefficient

Flow Similarity and Model Studies

• Scaling with Multiple Dependent Parameters

Example: Centrifugal Pump(Negligible Viscous Effects)

If … … then …

Flow Similarity and Model Studies

• Scaling with Multiple Dependent Parameters

Example: Centrifugal Pump

Specific Speed

• Recommended Text Book

• Frank M White Fluid Mechanics

• Mott, Applied Fluid Mechanics’, Sixth edition, Prentice Hall, 2006