# 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.

## advantages

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:

- frictional flows
- Currents with a free surface and suspended particles in currents
- Flows with pressure gradients in the main flow direction
- simultaneous heat and mass transfer
- the gas dynamics
- Heat transfer through convection in currents
- Heat transfer on a wall through flow

## List of key figures (parameters)

## Web links

- New dimensionless number in fluid mechanics (accessed on September 24, 2015)
- Manuscript for the lecture Heat Transfer (accessed September 24, 2015)
- Experimental and theoretical investigations to optimize the heat transfer in recuperative burners. (accessed on September 24, 2015)