NekRS can solve its equations in dimensional or non-dimensional form. When solving in non-dimensional form, characteristic scales for velocity, temperature, length, and time are defined and substituted into the Navier-Stokes equations so that all solution fields (velocity, pressure, temperature) are of order unity. Non-dimensional formulations for velocity, pressure, temperature, length, and time are defined as
ui†≡urefui(1)
P†≡ρ0uref2P(2)
T†≡ΔTT−Tref(3)
xi†≡Lrefxi(4)
t†≡Lref/ureft(5)
where a † superscript indicates a non-dimensional quantity and a ref subscript indicates a characteristic scale. A "0" subscript indicates a reference parameter (so-called because a reference parameter is not really a characteristic scale, but rather a reference value corresponding to the conditions at the characteristic scales). Inserting these definitions into the conservation of mass, momentum, and energy equations solved by NekRS gives
∂xi†∂ui†=0(6)
ρ†(∂t†∂ui†+uj†∂xj†∂ui†)=−∂xi†∂P†+Re1∂xj†∂τij†+ρ†fi†(7)
ρ†Cp†(∂t†∂T†+ui†∂xi†∂T†)=Pe1∂xi†∂(k†∂xi†∂T†)+q˙†(8)
New terms in these non-dimensional equations are Re and Pe, the Reynolds and Peclet numbers, respectively:
Re≡μ0ρ0urefLref(9)
Pe≡αLrefuref(10)
where α is the thermal diffusivity. NekRS solves for u†, P†, and T†. Cardinal will handle conversions from a non-dimensional NekRS solution to a dimensional MOOSE application.