Notes on Computational Fluid Dynamics: General Principles

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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=

(2.68)

with the dimensionless numbers²¹:

Strouhal number $\text{[math]}$ — transient/steady inertia;
Reynolds number $\text{[math]}$ — inertia/viscous force;
Euler number $\text{[math]}$ — pressure force/inertia;
(Froude number) $\text{[math]}$ $\text{[math]}$ — inertia/gravity force.

These dimensionless numbers include a characteristic length $\text{[math]}$ , time $\text{[math]}$ , speed $\text{[math]}$ and pressure $\text{[math]}$ . The $\text{[math]}$ (hat) notation indicates dimensionless length, time, etc., e.g. $\text{[math]}$ and $\text{[math]}$ and corresponding dimensionless operators $\text{[math]}$ and $\text{[math]}$ .

Equation 2.68 assumes constant $\text{[math]}$ and splits gravitational acceleration $\text{[math]}$ into its magnitude $\text{[math]}$ and unit direction $\text{[math]}$ ; pressure, including $\text{[math]}$ , is in kinematic units (divided by $\text{[math]}$ ).

The dimensionless numbers provide a comparison of the magnitudes of diﬀerent ﬂuid forces. For example, $\text{[math]}$ 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 $\text{[math]}$ , can be expressed in non-dimensional form as

@ ^T ^ ^ -1- ^2 ^ Sr @^t + r (^u T) Pe r T = 0; \relax \special {t4ht=

(2.69)

which includes the following additional dimensionless number:²²

Péclet number $\text{[math]}$ — advection/diﬀusion of heat.

Again in Eq. (2.69 ), the $\text{[math]}$ (hat) notation is applied to temperature $\text{[math]}$ to indicate a dimensionless temperature, although it notably does not appear in a dimensionless number.

In fact, with the exception of momentum (which uses $\text{[math]}$ ), $\text{[math]}$ 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 $\text{[math]}$ — viscosity/thermal diﬀusivity;
Schmidt number $\text{[math]}$ — viscosity/mass diﬀusivity.

where $\text{[math]}$ 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 $\text{[math]}$ .

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