# Outflow boundary condition - second derivative of velocity

Consider a fluid flow simulation in a pipe. At the outflow, instead of explicitly imposing a boundary condition, I linearly extrapolate information from the interior (for velocity components). This will translate to $$\frac{\partial ^2 u}{\partial x^2} =\frac{\partial ^2 v}{\partial x^2}=\frac{\partial ^2 w}{\partial x^2}= 0$$ at the boundary($x$ being axial direction). This kind of numerical condition has been used successfully throughout the literature (see page 262 of this paper).

What I am interested is - what does this condition physically mean?

Any discussion would be greatly appreciated.

• In your formulation is $v$ the velocity in the streamwise or flow normal direction? – Spencer Bryngelson Apr 19 '17 at 5:57
• @SpencerBryngelson : In my formulation (which is 3D), I set the second derivative of all components of velocity to be zero. – Rhinocerotidae Apr 19 '17 at 8:42
• This means, that the acceleration does not change in normal direction. But why the second derivative? Typically (and also in the reference you give), the velocity is prolonged over the boundary. – Jan Apr 20 '17 at 7:20
• @Jan : You mean 'convective acceleration'? The second derivative arises from the linear extrapolation. – Rhinocerotidae Apr 20 '17 at 9:48
• I haven't seen this used before. In a practical sense, why does a first derivative outflow condition not suffice? – Spencer Bryngelson Apr 21 '17 at 18:08