Non-Dimensional Formulation

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

uiuiuref u_i^\dagger\equiv\frac{u_i}{u_{ref}}(1)

PPρ0uref2 P^\dagger\equiv\frac{P}{\rho_0u_{ref}^2}(2)

TTTrefΔT T^\dagger\equiv\frac{T-T_{ref}}{\Delta T}(3)

xixiLref x_i^\dagger\equiv\frac{x_i}{L_{ref}}(4)

ttLref/uref t^\dagger\equiv\frac{t}{L_{ref}/u_{ref}}(5)

where a \dagger superscript indicates a non-dimensional quantity and a refref subscript indicates a characteristic scale. A "00" 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

uixi=0 \frac{\partial u_i^\dagger}{\partial x_i^\dagger}=0(6)

ρ(uit+ujuixj)=Pxi+1Reτijxj+ρfi \rho^\dagger\left(\frac{\partial u_i^\dagger}{\partial t^\dagger}+u_j^\dagger\frac{\partial u_i^\dagger}{\partial x_j^\dagger}\right)=-\frac{\partial P^\dagger}{\partial x_i^\dagger}+\frac{1}{Re}\frac{\partial \tau_{ij}^\dagger}{\partial x_j^\dagger}+\rho^\dagger f_i^\dagger(7)

ρCp(Tt+uiTxi)=1Pexi(kTxi)+q˙ \rho^\dagger C_p^\dagger\left(\frac{\partial T^\dagger}{\partial t^\dagger}+u_i^\dagger\frac{\partial T^\dagger}{\partial x_i^\dagger}\right)=\frac{1}{Pe}\frac{\partial}{\partial x_i^\dagger}\left(k^\dagger\frac{\partial T^\dagger}{\partial x_i^\dagger}\right)+\dot{q}^\dagger(8)

New terms in these non-dimensional equations are ReRe and PePe, the Reynolds and Peclet numbers, respectively:

Reρ0urefLrefμ0 Re\equiv\frac{\rho_0 u_{ref}L_{ref}}{\mu_0}(9)

PeLrefurefα Pe\equiv\frac{L_{ref}u_{ref}}{\alpha}(10)

where α\alpha is the thermal diffusivity. NekRS solves for u\mathbf u^\dagger, PP^\dagger, and TT^\dagger. Cardinal will handle conversions from a non-dimensional NekRS solution to a dimensional MOOSE application.