CMPS Theory Manual › Solution Monitoring and Integral Reports

Solution Monitoring and Integral Reports

This theory manual focuses on equations and numerics, but the project source also contains solver-level integral diagnostics that are part of how conservation and aerodynamic performance are assessed.

Surface fields and averages

For a surface \(S\), the area-weighted mean of a scalar \(\phi\) is

\[\langle\phi\rangle_A=\frac{\int_S\phi\,dA}{\int_SdA}.\]

When a mass-weighted report is requested, the weighting is based on signed or directed carrier mass flow as appropriate to the report definition,

\[\langle\phi\rangle_{\dot m}=\frac{\int_S\phi\,\rho\mathbf u\cdot\mathbf n\,dA}{\int_S\rho\mathbf u\cdot\mathbf n\,dA}.\]

Surface fields include density, pressure, temperature, total pressure, total temperature, velocity, Mach number, vorticity magnitude, enthalpy/energy and turbulence quantities.

Flux reports

Carrier mass flow through a surface is

\[\dot m=\int_S\rho\mathbf u\cdot\mathbf n\,dA.\]

The dilute dispersed-phase counterpart is

\[\dot m_d=\int_S\rho_d\mathbf V_d\cdot\mathbf n\,dA.\]

Total wall/surface heat rate follows

\[\dot Q=\int_S q''\,dA.\]

In axisymmetric calculations the surface-integration path applies the corresponding rotational geometric weighting.

Forces, moments and coefficients

The pressure and viscous forces are

\[\mathbf F_p=-\int_Sp\mathbf n\,dA,\qquad \mathbf F_\tau=\int_S\boldsymbol\tau\cdot\mathbf n\,dA,\qquad \mathbf F=\mathbf F_p+\mathbf F_\tau.\]

Moment about reference point \(\mathbf x_0\) is

\[\mathbf M_{x_0}=\int_S(\mathbf x-\mathbf x_0)\times d\mathbf F.\]

With dynamic pressure \(q_{ref}=\tfrac12\rho_{ref}U_{ref}^2\),

\[C_D=\frac{\mathbf F\cdot\mathbf e_D}{q_{ref}A_{ref}},\qquad C_L=\frac{\mathbf F\cdot\mathbf e_L}{q_{ref}A_{ref}}.\]

The reported center of pressure is derived from the force/moment balance relative to the selected reference geometry.

Point probes and convergence monitoring

Point probes and iterative monitors provide local solution histories, residual evolution, time-step/CFL information and solver status. These diagnostics do not alter the governing residual; they sample or reduce the distributed solution state.