ME2342_Sec5_DimensionalAnalysis

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P. S. Krueger ME/CEE 2342 5 - 1

ME/CEE 2342:

Paul S. Krueger

Associate Professor

Department of Mechanical Engineering

Southern Methodist UniversityDallas, TX 75275

[email protected]

(214) 768-1296Office: 301G Embrey

Fluid Mechanics

Section 5 – Dimensional Analysis

[Chapter 7 in the text book]

mailto:[email protected]

mailto:[email protected]




Venturi Tube:

Objective: Find the flow rate Q as a function of Δ p and theother parameters in the problem, namely,

( )ρΔ= ,,, 21 A A p f Q




Using Bernoulli’s equation (μ ignored) and conservation of

mass we found (see previous notes)

or

Could we have guessed the functional dependence in

advance? Yes! – Using a theorem called …


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( )nvvvV v ,,, 321K=

The Buckingham Pi Theorem:

If there is a functional relationship involving n variables,

Then it can be reduced to a relationship among k = n – jindependent dimensionless parameters

( )k

V ΠΠΠ=Π ,,,~

321

K ( ) jnk −=

where j is the minimum number of reference dimensions

(such as mass, length, time) required to describe the

dimensions of all the variables.



Example: Venturi Tube

Specifically,



• The principle behind the Pi theorem is that only so many

independent combinations of variables can be formedand one of these dimensionless parameters must be a

function of the others.

• The functional relationship between Pi terms isdetermined by physics (e.g., Bernoulli’s equation) or

experiment. The advantage of the theorem is that is

reduces the number of independent variables involved inthe analysis of a problem.

• Pi terms are independent, but not unique.

Notes:



Using the Pi Theorem: Method of Repeating Variables

Section 7-4 in the book gives 6 steps for determining the Piterms.

Example: Shear force to move a plate

Step 1: List the variables involved




Step 2: List dimensions of the terms

Step 3: Determine the number of required Pi terms

Step 4: Select repeating variables. That is, select j variables

that contain the reference dimensions and use these to non-

dimensionalize the remaining variables. Note: do not selectthe dependent variable as a repeating variable.




Step 5: Find the Pi terms by non-dimensionalizing the

remaining variables using the repeating variables.




Step 6: Check Pi terms

Conclude (‘Step 7’) by writing down the relationship among

Pi terms:




• We can rewrite Π4:

• Even though we don’t know the functional relationship,

we still have a lot of information…

Notes:

• We could have non-dimensionalized F D

as




• Selection of Variables

[See section 7-3 in the textbook]

Notice that the key to dimensional analysis is knowing which

variables to include in the analysis. It isn’t always clear,

however, which variables we should choose to include. If we choose too many, we over-complicate the analysis. If we

choose too few, we may miss a key behavior. How do we

know which variables to include in our analysis?

- Experience

- Basic physical insight (knowledge of physics involved)




Example: Pendulum Bob

What parameters are involved?




What about aerodynamic drag?

What about θ?

What about moment of inertia I ?




Based on our observations so far, we expect

Pi term “by inspection”:

Note: mass is not involved. Dimensionless relationship:




• Common Dimensionless Parameters

[See also Table 7-5]

1) Reynolds Number:

For a physical interpretation of Reynolds number, consider the following:

Inertia “forces”/terms:

μ

ρ=

UL Re




Then:

Viscous Forces:

So,




Physical Interpretation:

Re >> 1: Inertia effects (i.e., effects associated withacceleration or deceleration of fluid particles) dominate.

That is, it is difficult for viscosity to slow the fluid (flow has a

tendency to keep moving by inertia). Viscous effects

typically confined to a small region of the flow.

Examples:

Re << 1: Viscous forces dominate everywhere in the flow. It

is difficult to keep the fluid moving (viscosity kills the motion).

Examples:

2) M h N b




2) Mach Number: cU Ma =


Ma << 1: Compressibility not important (i.e., assume

incompressible flow). If Ma ≤ 0.3 we say flow is

incompressible.

Example:

Ma >> 1: Compressibility important.

3) St h l N b (di i l f )




3) Strouhal Number (dimensionless frequency):

U

L

St

ω

=


St << 1: Unsteadiness not important relative to convective

effects.

St > 1: Unsteadiness important.

4) Euler Number (dimensionless pressure):

2U

p

Eu ρ= or 22U

p

Eu ρ=

Used in problems where pressure differences are important

(e.g, Bernoulli’s equation)

4) F d N b




4) Froude Number:

Important in problems with surface waves.

gL

U Fr =

or gL

U

Fr

2

=

Th f M d l




• Theory of Models

[Section 7-3 (Dimensional analysis and similarity)]

Consider a case where we know a relationship exists

between parameters for a problem:

Then using dimensional analysis we arrive at

But the functional relationship is unknown and needs to be

determined by experiment. For cost and convenience, thisis often done using a scale model.

P t t M d l




Prototype

(full-scale system)

Model

(scaled system)

Thus, if we match all the independent Pi terms, we can

predict the dependent Pi term for the full-scale system

N t




• Matching Pi terms involving only geometric parameters

(length, shape, etc.) gives geometric similarity. Inparticular, the model must be a scale model of the

prototype (identical geometry to the prototype, except,

possibly, a different size).• Matching Pi terms involving ratios of forces enforces

dynamic similarity. For example, if Reynolds number

is a Pi term, then Rem = Re p is required for dynamicsimilarity. Enforcing dynamic similarity ensures the flow

around the model is the same as the flow around the

prototype, just at a different scale.

Notes:

Example: Drag on an Airplane




Example: Drag on an Airplane

What static pressure in the wind tunnel is required to

achieve dynamic similarity? If the measured drag on the

model is F Dm = 1 lbf , what would be the drag on the

prototype?

First we need to apply dimensional analysis to the




First, we need to apply dimensional analysis to the

relationship for drag on the airplane:

Result:

So the only Pi term that needs to be matched is Re Also




So, the only Pi term that needs to be matched is Re. Also,

the only Pi term involving ratios of forces is Re, so dynamic

similarity requires Re p = Rem, namely,

AssumeThen dynamic similarity gives

Now use the ideal gas law:

Drag: When Re = Re dimensional analysis also requires




Drag: When Re p = Rem dimensional analysis also requires

that

Note: It isn’t always possible to achieve full dynamic




Note: It isn t always possible to achieve full dynamic

similarity with scale models. For example, for model ships

the flow involves a free surface and both Re and Fr areimportant. Then full dynamic similarity requires

It is nearly impossible to find two fluids with appropriate ν /ν




It is nearly impossible to find two fluids with appropriate νm/ν pfor reasonable scales. For example,

Instead, ship designers typically utilize Re independence

when testing models. Here the strategy is to match Fr for the model and prototype, but keep the model large enough

so that Re independence is achieved. Under Re

independence, the Re is large enough that the drag (or other fluid dynamic quantity) no longer depends on Re. Then,

even though Rem < Re p, the value(s) at Rem can still be used

to estimate the prototype quantities. See the discussion in

the textbook for a more complete analysis.

ME2342_Sec5_DimensionalAnalysis

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