Dimensional analysis

CHEE 3363Spring 2014Handout 17

�Reading: Fox, 7.3–7.5

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Learning objectives for lecture

1. Apply the Buckingham Pi Theorem to determine functional dependence on given parameters.�

2. Derive the dimensionless form of the Navier-Stokes equations.�

3.and state their physical meaning.�

4. Contrast kinematic similarity, dynamic similarity, and geometric similarity.

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Dimensional analysisWhat kind of questions can we answer using dimensional analysis?�

Procedure: Buckingham Pi Theorem

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Π parametersGiven: a problem in which the dependent parameter is a function of n - 1 independent parameters:Need: n - m dimensionless ratios, where m is (usually) equal to the minimum number of independent dimensions required to specify the dimensions of all the parameters.

1. List all dimensional parameters involved�

2. Select a set of fundamental (primary) dimensions�

3. List the dimensions of all parameters in terms of primary dimensions�

4. Select a set of r dimensional parameters that includes all the primary dimensions (do not include dependent parameter)�

5. Set up dimensional equations, combining parameters in Step 4 with each of other parameters in turn to form dimensionless groups�

6. Check to see that each group obtained is dimensionless�4

Swinging pendulum via Buckingham Pi 1

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Given: the period T is a function of pendulum length L, bob mass m, and gravity g�Find: the dimensionless group(s)

1. List all dimensional parameters involved

Recall from handout 1: T =

sL

g

2. Select a set of fundamental primary dimensions

Swinging pendulum via Buckingham Pi 2

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Given: the period T is a function of pendulum length L, bob mass m, and gravity g�Find: the dimensionless group(s)3. List dimensions of all parameters in terms of primary dimensions

4. Select a set of r dimensional parameters that includes all the primary dimensions (but do not include dependent parameter)

Swinging pendulum via Buckingham Pi 3

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Given: the period T is a function of pendulum length L, bob mass m, and gravity g�Find: the dimensionless group(s)5. Set up dimensional equations, combining parameters in Step 4 with each of other parameters in turn to form dimensionless groups

Swinging pendulum via Buckingham Pi 4

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Given: the period T is a function of pendulum length L, bob mass m, and gravity g�Find: the dimensionless group(s)6. Check to see that each group obtained is dimensionless

Application to Navier-Stokes

The Buckingham Pi theorem is a procedure that allows us to identify dimensionless groups�

- This allows us to determine the important combination of parameters for an experimental system without having to do every experiment!�

We will now apply dimensional analysis to non-dimensionalize the Navier-Stokes equations and gain further physical insight

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F = f(D,V, ρ, µ)

F

ρV 2D2= f

(

ρV D

µ

)

Dimensional analysis motivationGoal

e.g. consider drag force on sphere

diameter

velocitydensity

viscosity

Testing complete dependence of drag force on all parameters would take an extremely long time

Instead, we will show that we can describe the problem completely with two dimensionless parameters:

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Example: drag force on a sphere 11. List all dimensional parameters involved

2. Select a set of fundamental primary dimensions

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3. List dimensions of all parameters in terms of primary dimensions

Example: drag force on a sphere 2

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4. Select a set of r dimensional parameters that includes all the primary dimensions (but do not include dependent parameter)

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

Example: drag force on a sphere 3

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

Example: drag force on a sphere 4

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Note that dimensional analysis simply tells you what combination of parameters appears in the solution; in particular, the functional form of f must still be determined by solving!

Also note: the choice of {ρ,V,D} as repeating parameters is generally a good choice, because inertial forces are important in most fluids problems

6. Check to see that each group obtained is dimensionless

Non-dimensionalizing the equations 1

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Consider

Continuity:

Navier-Stokes:

x:

y:

Coupled nonlinear PDEs for u, v, p

Units of each equation:�Continuity: �N-S:

Goal: put all equations in dimensionless form (i.e. no units)

∂u

∂x+

∂v

∂y= 0

ρ

(

u∂u

∂x+ v

∂u

∂y

)

= −

∂p

∂x+ µ

(

∂2u

∂x2+

∂2u

∂y2

)

ρ

(

u∂v

∂x+ v

∂v

∂y

)

= −ρg −

∂p

∂y+ µ

(

∂2v

∂x2+

∂2v

∂y2

)

Non-dimensionalizing the equations 2

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Procedure: divide lengths by a characteristic length, and velocities by a characteristic velocity

Note: we also need to non-dimensionalize the pressure -- what to use?

[Pressure] = [force / area] = [M/ LT2] = [(M/L3)(L2/T2)] = [density velocity2]

Substitute terms into the equations:

Non-dimensionalizing the equations 3

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Substitute terms into the equations:

Result:Continuity:

Navier-Stokes:

x:

y:

Non-dimensionalizing the equations 4

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Divide continuity equation by V / L and N-S by ρV2 / L:

Continuity:

Navier-Stokes:

x:

y:

What we have done: shown that solutions depend on two dimensionless parameters:

Magnitude of these parameters tells us whether gravity or viscosity is important — useful insight into problem even before solving!

If viscosity is small:�

�

�

�

�

in thin boundary layer, will always have an effect of viscosity�

- Related to the fact that we lose a boundary condition by going from 2nd order to 1st order equation�

Non-dimensional forms of equations allow us to determine which physical forces are dominant in a problem

Comments on non-dimensional eqns

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Inertial force

Viscous force

Pressure force

Gravity force

Surface tension force

Compressibility force

How do we determine which of these forces are more important?Compare magnitudes of forces using dimensionless parameters

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=ρV 2L2

µV L=

ρV L

µReynolds number Re =

inertial forceviscous force

=∆pL2

ρV 2L2≈

∆p1

2ρV 2

Euler number Eu = pressure forceinertial force

(factor of ½ introduced so that denominator is dynamic pressure)p − pvapor

1

2ρV 2

Cavitation number Ca =

=ρV 2L2

gρL3=

V 2

gLFroude number Fr2 =

inertial forcegravity force

(square is for historical reasons)

=ρV 2L2

σL=

ρV 2L

σWeber number We =

inertial forcesurface tension force

=ρV 2L2

EvL2=

V 2

Ev/ρ=

V 2

c2Mach number M2 =

inertial forcecompressibility force

Comments on similarity analysis

phenomena on a full-scale prototype:�

- Geometric similarity: �

- Kinematic similarity: �

- Dynamic similarity:�

Kinematic similarity requires geometric similarity (bounding streamlines also must scale)�

for dynamic similarity �22