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I am solving an unsteady flow using the dual-time Navier-Stokes equation in which I write my momentum equation as: $$\frac{\partial u}{\partial \tau} + \frac{\partial u}{\partial t} + \frac{\partial u^2}{\partial x} + \frac{\partial uv}{\partial y} = -\frac{\partial p}{\partial x} + \frac{\partial^2 u}{\partial x ^2} + \frac{\partial^2 u}{\partial y ^2}$$

If the time levels in 'physical time, t' goes like n-1, n, n+1, and for that of 'pseudo time, $\tau$ goes like q-1, q, q+1. Then, with the combination of forward and central difference, I can discretize this equation as:

$$\frac{u^{q+1}-u^{q}}{d\tau} + \frac{3u^{q+1}-4u^{n}+u^{n-1}}{2dt} + C = -P + D$$

where C, P and D are the discretized version of convection, pressure and diffusion terms respectively. I use a forward difference for pressure, central difference for diffusion terms.

Iterations are done in pseudo time unless the local steady state is achieved on which I simply assign $u^{n+1} = u^{q+1}$. My concern is of stability at this moment. I seem to infer that both dt and $\tau$ affects the stability but I can not come to a mathematical expression. Therefore I am looking forward to gain some knowledge about how to calculate the CFL number (viscous and advection) for this particular scheme.

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It is done in the standard way through von Neumann stability analysis. The CFL condition on the pseudo timestep is straight forward. The restrictions due to the viscosity and other terms are a little complicated, but they can still be obtained.

In the following paper the authors workout the time-step restriction on the pseudo time for an explicit Euler scheme for Artificial Compressibility and backward Euler scheme for the real time (in place of the BDF2 that you have). They show how the pseudo timestep is affected by the real timestep, viscosity and the AC parameters.

McHugh, P. R., and John D. Ramshaw. "Damped artificial compressibility iteration scheme for implicit calculations of unsteady incompressible flow." International journal for numerical methods in fluids 21.2 (1995): 141-153.

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