# Dimensionless key figure

A dimensionless number , similarity number or parameter is a parameter in a dimensionless mathematical model of a physical state or process. If two states or processes are defined by the same mathematical model, all quantities of one can be converted into those of the other with a given transformation rule if and only if the dimensionless key figures have the same values. Both processes or states are then similar to one another. Dimensionless key figures usually result from a de-dimensionalization of the mathematical model.

The advantage of the dimensionless key figures lies in the possibility of using a few exemplary measurements in the model test to determine the solution for any other cases in which the dimensionless key figures are the same as in the model test.

## application areas

Dimensionless key figures or quantities of the dimension number characterize physical processes that result from the similarity theory or dimension analysis .

The main area of ​​application for dimensionless indicators in technical mechanics is called similarity mechanics (→ Buckingham's Π-theorem , dimensional analysis ):

### Dimension analysis formula

The number of measured variables involved minus the number of basic units contained (basic dimensions) results in the number of key figures (dimensionless groups).

In fluid dynamics , for example, the flow around a body is described using the Navier-Stokes equation in conjunction with the continuity equation and boundary conditions (geometry of the body and other limitations). The coefficients of the dimensionless Navier-Stokes equation are the Reynolds number , Froude number and, in the unsteady case, the Keulegan-Carpenter number .

The Froude number has an influence on problems with a free surface, so it is relevant in shipbuilding and offshore engineering, and describes, for example, how long a ship is compared to waves that propagate at the same speed as the ship is traveling. The Reynolds number describes the effect of viscosity . The Keulegan-Carpenter number can, for example, describe in a dimensionless manner what effect the sea has on offshore structures.

example

If, for example, the drag and dynamic lift per length have been measured for a series of Reynolds numbers and angles of attack on a certain profile on a reduced scale, the results can be converted to profiles of any size of the same cross-sectional shape by ensuring that the Reynolds Number is the same as when measuring.

Shipbuilding research institutes make a living from simulating the flow around moving ships on a model scale and should actually simulate both the Reynolds number and the Froude number of the ship. Because this is not possible as long as one does not run huge models on a scale of 1: 4 in mercury instead of water, one restricts oneself to observing the Froude number and corrects the measurement results empirically by taking the frictional resistance from the Reynolds number of the model converted to that of the large version.

### Further areas of application

We know key figures in:

## List of key figures (parameters)

Surname Formula symbol annotation
Abbe number ν
Archimedes number Ar
Arrhenius number γ
Atwood number At
Lift coefficient c a
Fumigation number Q
Biot number Bi
Bodenstein number Bo
Bond number Bo
Brinkmann number Br
Bulk Richardson Number
Cauchy number Approx
Colburn number J
Courant number Co
Damköhler number There
Dean number De
Deborah number De
Pressure loss coefficient ζ
Printing number ψ
Flow rate φ
Eckert number Ec
Ekman number Ek
Alsatian number
Eötvös number Eo
Ericksen number He
Euler number Eu
Fourier number Fo
Froude number Fr.
Galileo number Ga
Graetz number Gz
Grass yard number Size
Hagen number Ed
Hartmann number Ha
Hatta number Ha
Helmholtz number Hey
Jacob number Yes
Capillary number
Karlovitz number Ka
Cavitation number σ
Keulegan-Carpenter number KC
Knudsen number Kn
Laplace number La
Run number σ
Laval number M *
Performance figure λ
Lewis number Le
Lyascenko number Lj Omega number
Mach number Ma
Marangoni number Mg
Landmark Number
Morton number Mon
Taking number N / A also name Griffith number
Newton number No
Nusselt number Nu
Uncare number Oh
Péclet number Pe
Phase transition number Ph Reciprocal of the Stefan number
Prater number β
Prandtl number Pr
Rayleigh number Ra
Reynolds number re
Richardson number
Pipe friction coefficient λ
Rossby number Ro
Schmidt number Sc
High speed number λ
Sherwood number Sh
Boiling number Bo according to boiling number
Sommerfeld number So
Stanton number St.
Stefan number Ste Reciprocal of the phase transition number
Stokes number St.
Strouhal number Sr
Drag coefficient c w
Suratman number
Taylor number Ta
Thiele module Φ
Weber number We
Weisz module Ψ
Weissenberg number Ws