System of Navier-Stokes equations

TRACE solves the compressible three-dimensional Navier-Stokes equations in a rotating frame of reference. In differential form these equations may be written as follows

\begin{equation} \frac{\partial \mathbf{\hat{Q}}}{\partial t} + \frac{\partial \mathbf{\hat{F}}}{\partial \xi} + \frac{\partial \mathbf{\hat{G}}}{\partial \eta} + \frac{\partial \mathbf{\hat{H}}}{\partial \zeta} = \left[\frac{\partial \mathbf{\hat{F}}_v}{\partial \xi} + \frac{\partial \mathbf{\hat{G}}_v}{\partial \eta} + \frac{\partial \mathbf{\hat{H}}_v}{\partial \zeta}\right] + \mathbf{\hat{S}} \end{equation}

where \(\mathbf{\hat{Q}} = (1/J)\left[\rho, \rho u, \rho v, \rho w, \rho E\right]^T\) denotes the solution vector and \(J = \partial(\xi,\eta,\zeta,t)/\partial(x,y,z,t)\) is the transformation Jacobian. In TRACE, as the rotation axis coincides with the \(x\)-coordinate axis, the angular velocity has the components \(\mathbf{\Omega} = \left[\omega, 0, 0\right]^T\). The effective Coriolis and centrifugal forces arising from this motion are contained in the source term \(\mathbf{\hat{S}}\). In the absence of gravity and other body forces this is defined as follows

\begin{equation} \mathbf{\hat{S}} = \frac{1}{J}\left[ \begin{array}{c} 0 \\ 0 \\ \rho \omega (y \omega + 2w) \\ \rho \omega (z \omega - 2v) \\ 0 \end{array} \right] \end{equation}

where \(y\) and \(z\) are the Cartesian coordinates in the \(y\)- and \(z\)-directions, respectively. The inviscid \(\mathbf{\hat{F}}\), \(\mathbf{\hat{G}}\), \(\mathbf{\hat{H}}\) and viscous fluxes \(\mathbf{\hat{F}}_v\), \(\mathbf{\hat{G}}_v\), \(\mathbf{\hat{H}}_v\) are

\begin{equation} \mathbf{\hat{F}} = \frac{1}{J}\left[ \begin{array}{c} \rho U \\ \rho U u + \xi_x p \\ \rho U v + \xi_y p \\ \rho U w + \xi_z p \\ (\rho E + p)U - \xi_t p \end{array} \right] \quad , \quad \mathbf{\hat{F}}_v = \frac{1}{J}\left[ \begin{array}{c} 0 \\ \xi_x \tau_{xx} + \xi_y \tau_{xy} + \xi_z \tau_{xz} \\ \xi_x \tau_{xy} + \xi_y \tau_{yy} + \xi_z \tau_{yz} \\ \xi_x \tau_{xz} + \xi_y \tau_{yz} + \xi_z \tau_{zz} \\ \xi_x b_x + \xi_y b_y + \xi_z b_z \end{array} \right] \end{equation}

\begin{equation} \mathbf{\hat{G}} = \frac{1}{J}\left[ \begin{array}{c} \rho V \\ \rho V u + \eta_x p \\ \rho V v + \eta_y p \\ \rho V w + \eta_z p \\ (\rho E + p)V - \eta_t p \end{array} \right] \quad , \quad \mathbf{\hat{G}}_v = \frac{1}{J}\left[ \begin{array}{c} 0 \\ \eta_x \tau_{xx} + \eta_y \tau_{xy} + \eta_z \tau_{xz} \\ \eta_x \tau_{xy} + \eta_y \tau_{yy} + \eta_z \tau_{yz} \\ \eta_x \tau_{xz} + \eta_y \tau_{yz} + \eta_z \tau_{zz} \\ \eta_x b_x + \eta_y b_y + \eta_z b_z \end{array} \right] \end{equation}

\begin{equation} \mathbf{\hat{H}} = \frac{1}{J}\left[ \begin{array}{c} \rho W \\ \rho W u + \zeta_x p \\ \rho W v + \zeta_y p \\ \rho W w + \zeta_z p \\ (\rho E + p)W - \zeta_t p \end{array} \right] \quad , \quad \mathbf{\hat{H}}_v = \frac{1}{J}\left[ \begin{array}{c} 0 \\ \zeta_x \tau_{xx} + \zeta_y \tau_{xy} + \zeta_z \tau_{xz} \\ \zeta_x \tau_{xy} + \zeta_y \tau_{yy} + \zeta_z \tau_{yz} \\ \zeta_x \tau_{xz} + \zeta_y \tau_{yz} + \zeta_z \tau_{zz} \\ \zeta_x b_x + \zeta_y b_y + \zeta_z b_z \end{array} \right] \end{equation}

where \(\rho\), \(u\), \(v\), \(w\), \(p\) and \(E\) are the instantaneous density, Cartesian velocity components, static pressure and specific relative total energy, respectively. The specific relative total energy is defined as

\begin{equation} E = \varepsilon + \frac{1}{2}\left(u^2 + v^2 + w^2\right) - \frac{1}{2}\omega^2_1 r^2 \end{equation}

where \(\varepsilon\) is the specific internal energy density and \(r = \sqrt{y^2 + z^2}\). The pressure is related to the other thermodynamic variables by the used gas model (see Gas models) The terms \(U\), \(V\) and \(W\) in the inviscid fluxes are the contravariant velocities. These are defined as follows

\begin{eqnarray} U &=& \xi_x u + \xi_y v + \xi_z w + \xi_t \\ V &=& \eta_x u + \eta_y v + \eta_z w + \eta_t \\ W &=& \zeta_x u + \zeta_y v + \zeta_z w + \zeta_t \end{eqnarray}

The terms \(\tau_{ij}\) are the viscous stresses. These are related to the Cartesian velocity components by the equation

\begin{equation} \tau_{ij} = \mu\left[\left(\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j}\right) - \frac{2}{3}\frac{\partial u_k}{\partial x_k}\delta_{ij}\right] \end{equation}

where \(\mu\) is the dynamic molecular viscosity and \(\delta_{ij}\) is the Kronecker delta. The dynamic molecular viscosity is related to temperature using Sutherland's law for calorically perfect gas. The terms \(b_j\) are defined as follows

\begin{equation} b_j = u_i\tau_{ij} + \frac{1}{(\gamma - 1)}\mu\frac{\partial T}{\partial x_j}. \end{equation}

Continuity, momentum and energy equation

The System of Navier-Stokes equations can be divided into mass, momentum and energy conservation equations. Following Einstein's convention the equations read in differential conservation form:

\begin{eqnarray} \text{continuity} \qquad \qquad \qquad \qquad \qquad \qquad \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_i}\left(\rho u_i\right) &=& 0 \\ \text{momentum} \qquad \qquad \qquad \qquad \quad \frac{\partial}{\partial t}\left(\rho u_i\right) + \frac{\partial}{\partial x_j}\left(\rho u_j u_i\right) &=& - \frac{\partial p}{\partial x_i} + \frac{\partial \tau_{ij}}{\partial x_j} \\ \text{energy} \quad \frac{\partial}{\partial t} \left[\rho \left(\epsilon + \frac{u_i u_j}{2}\right)\right] + \frac{\partial}{\partial x_j} \left[u_j \rho \left(h+\frac{u_i u_j}{2}\right)\right] &=& \frac{\partial}{\partial x_j}\left[-q_j + u_i \tau_{ij}\right] \end{eqnarray}

where \(q_j\) is the molecular heat flux vector.