2.21 Scale similarity

Scale similarity is the notion that, for two systems that are geometrically similar, the flow will follow the same path if the ratio of magnitude of forces acting on the fluid is the same at different points in the flow.

Flows at different scales can be compared using dimensionless variables. The momentum equation, Eq. (2.67 ), with advection, diffusion and gravity forces can be expressed in non-dimensionalised form by

 Sr @^u-+ ^r (^u^u) 1-- ^r2u^= @^t Re ^ -1- Eu r^p + Fr2 ng ^; \relax \special {t4ht=
with the dimensionless numbers21:
  • Strouhal number eqn — transient/steady inertia;
  • Reynolds number eqn — inertia/viscous force;
  • Euler number eqn — pressure force/inertia;
  • (Froude number)eqn eqn — inertia/gravity force.

These dimensionless numbers include a characteristic length eqn, time eqn, speed eqn and pressure eqn. The eqn (hat) notation indicates dimensionless length, time, etc., e.g. eqn and eqn and corresponding dimensionless operators eqn and eqn .

Equation 2.68 assumes constant eqn and splits gravitational acceleration eqn into its magnitude eqn and unit direction eqn; pressure, including eqn, is in kinematic units (divided by eqn).

The dimensionless numbers provide a comparison of the magnitudes of different fluid forces. For example, eqn represents the ratio of inertia force to viscous force and plays a pivotal role in turbulence modelling, introduced in Chapter 6 .

Scale similarity applies also to other transported properties. For example, the energy equation, Eq. (2.65 ), ignoring heat sources eqn, can be expressed in non-dimensional form as

 @ ^T ^ ^ -1- ^2 ^ Sr @^t + r (^u T) Pe r T = 0; \relax \special {t4ht=
which includes the following additional dimensionless number:22
  • Péclet number eqn — advection/diffusion of heat.

Again in Eq. (2.69 ), the eqn (hat) notation is applied to temperature eqn to indicate a dimensionless temperature, although it notably does not appear in a dimensionless number.

In fact, with the exception of momentum (which uses eqn), eqn represents more generally the rates of advection and diffusion, as a ratio, for any transported quantity (here, it is heat).

Further numbers23 define the ratios of diffusivities, e.g.:

  • Prandtl number eqn — viscosity/thermal diffusivity;
  • Schmidt number eqn — viscosity/mass diffusivity.

where eqn is mass diffusivity (not discussed in this book).

Dimensionless numbers can be multiplied and divided with one another to form further dimensionless numbers. For example, the Péclet number for heat transfer eqn.

21After Vincenc Strouhal (1850-1922), Osborne Reynolds (1842-1912), Leonhard Euler (1707-1783) and William Froude (1810-1879).
22After Jean Claude Eugène Péclet (1793-1857).
23After Ludwig Prandtl (1875-1953) and Ernst Heinrich Wilhelm Schmidt (1892-1975).

Notes on CFD: General Principles - 2.21 Scale similarity