  In this paper, we propose two techniques to estimate the magnitude of a
machine-zero residual for a given problem, which is the smallest possible
residual that can be achieved when we solve a system of discretized equations.
We estimate the magnitude of the machine-zero residual by a norm of residuals
computed with a randomly-perturbed approximate solution that is considered as
close in magnitude to an exactly-converged solution. One method uses
free-stream values as the approximate solution, and the other uses a current
solution during an iterative solve as the approximate solution via the method
of manufactured solutions. Numerical results show that these estimates predict
the levels of machine-zero residuals very accurately for all equations of the
Euler and Navier-Stokes equations in a transonic flow over an airfoil and
viscous flows over a cylinder and a flat plate.
