Dimensional analysis 1

If down-scaled - how can we interpret the results?

What size?
Same tests? Same conditions?
Important variables?
How can we use known dimensionless relationships to predict results?

Steps in practise

List important variables
Example: flow around airplane
- Resistance (drag)
- Lift force
- Velocity
- …

Let’s focus on drag

example with cylinder

$Reynolds: R e = \frac{ρ UL}{μ}$ $Drag coefficient: C_{D} = \frac{F _{D}}{\frac{1}{2} ρ A v ^{2}}$

Similarities: (Between model and prototype)

Geometrical similarity:
- $\frac{L _{p}}{L _{m}} = \frac{h _{p}}{h _{m}} = constant$
- $L_{r} = \frac{L _{p}}{L _{m}}$
Kinematic similarity:
- Similar geometrical flows and velocities
- $V_{r} = \frac{V _{p}}{V _{m}}$
Dynamic similarity:
- Same ratios between forces in model and prototype
- $F_{r} = \frac{F _{p}}{F _{m}}$

Variables:

Drag force $F_{D}$ $M L T^{- 2}$
Speed $U$ $L T^{- 1}$
Length $L$ $L$
Density of the fluid $ρ$ $M L^{- 3}$
Dynamic viscosity $μ$ $M L^{- 1} T^{- 1}$

Number of $π$ -groups: $5 Variables - 3 Dimensions = 2 π -groups$

Choose repeating variables:

Speed
Length
Density

$Π_{1} = F_{D} U^{a_{1}} ρ^{b_{1}} d^{c_{1}}$

$[M^{0} L^{0} T^{0}] = [M L T^{- 2}] [L T^{- 1}]^{a_{1}} [M L^{- 3}]^{b_{1}} [L]^{c_{1}}$

$Π_{2} = μ U^{a_{2}} ρ^{b_{2}} d^{c_{2}}$

$[M^{0} L^{0} T^{0}] = [M L^{- 1} T^{- 1}] [L T^{- 1}]^{a_{2}} [M L^{- 3}]^{b_{2}} [L]^{c_{2}}$

Alternative route:

Compare list of variables to list of known dimensionaless variables and pick from there

Example problem B.1

$T = 20° C$ $ρ = 1.0 bar$ $U = 90 km/h$

Drag force $F_{D}$ $M L T^{- 2}$
Speed $U$ $L T^{- 1}$
Length $L$ $L$
Density of the fluid $ρ$ $M L^{- 3}$
Dynamic viscosity $μ$ $M L^{- 1} T^{- 1}$

Out: $Reynolds: R e = \frac{ρ UL}{μ}$ $Drag coefficient: C_{D} = \frac{F _{D}}{\frac{1}{2} ρ A v ^{2}}$

$R e = \frac{ρ UL}{μ}$

Similarity: $R e_{m} = R e_{p}$

$ρ_{m} = ρ_{p} = ρ$

$μ_{m} = μ_{p} = μ$

$\Rightarrow U_{m} = \frac{U _{p} L _{p}}{L _{m}}$

Model Scale 1:3

$\Rightarrow L_{m} = \frac{L _{p}}{3}$

$U_{p} = 90 km/h = \frac{90000}{3600} = 25 m/s$

$\Rightarrow U_{p} = \frac{U _{p} L _{p}}{L _{m}} = 3 U_{p} = 75 m/s = 270 km/h$

$C_{D} = \frac{F _{D}}{\frac{1}{2} ρ U ^{2} A}$

Similarity: $C_{D m} = C_{D p}$

$\frac{F _{D m}}{\frac{1}{2} ρ U _{m}^{2} A _{m}} = \frac{F _{D p}}{\frac{1}{2} ρ U _{p}^{2} A _{p}}$

$\Rightarrow F_{D p} = \frac{U _{p}^{2} A _{p} F _{D m}}{U _{m}^{2} A _{m}}$

$A = L^{2}$

$L_{m} = \frac{L _{p}}{3}$

$\Rightarrow A_{m} = (\frac{L _{p}}{3})^{2} = \frac{A _{p}}{9}$

$U_{m} = 3 U_{p} \Rightarrow U_{m}^{2} = 9 U_{p}$

$\Rightarrow F_{D p} = \frac{U _{p}^{2} A _{p} F _{D m}}{9 U _{p}^{2} \frac{A _{p}}{9}} = F_{D m} = 150 N$

🪴 n4kruG's Notes

Explorer

Dimensional analysis 1

Alternative route:

Example problem B.1

Example problem B.6

Graph View

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