Incompressible Navier-Stokes Equations

The incompressible Navier-Stokes equations solved in NekRS are conservation of mass,

uixi=0 \frac{\partial u_i}{\partial x_i}=0(1)

conservation of momentum,

ρf(uit+ujuixj)=Pxi+τijxj+ρfj \rho_f\left(\frac{\partial u_i}{\partial t}+u_j\frac{\partial u_i}{\partial x_j}\right)=-\frac{\partial P}{\partial x_i}+\frac{\partial \tau_{ij}}{\partial x_j}+\rho f_j(2)

and conservation of energy,

ρfCp,f(Tft+uiTfxi)=xi(kfTfxi)+q˙f \rho_f C_{p,f}\left(\frac{\partial T_f}{\partial t}+u_i\frac{\partial T_f}{\partial x_i}\right)=\frac{\partial}{\partial x_i}\left(k_f\frac{\partial T_f}{\partial x_i}\right)+\dot{q}_f(3)

where u\vec{u} is the velocity, PP is the pressure, ρf\rho_f is the fluid density, TfT_f is the fluid temperature, τ\tau is the viscous stress tensor, f\vec{f} is a force vector, q˙f\dot{q}_f is a volumetric heat source in the fluid, Cp,fC_{p,f} is the fluid isobaric specific heat capacity, and kfk_f is the fluid thermal conductivity.

Passive Scalars

A passive scalar is any quantity ss which satisfies an equation of the form

a(st+uisxi)=xi(bsxi)+q˙a\left(\frac{\partial s}{\partial t}+u_i\frac{\partial s}{\partial x_i}\right)=\frac{\partial}{\partial x_i}\left(b\frac{\partial s}{\partial x_i}\right)+\dot{q}

where aa and bb are coefficients. For instance, temperature is a passive scalar if a=ρCpa=\rho C_p and b=kfb=k_f. Other quantities which obey passive scalar equations include many aspects of mass transport.