# why is the third flux term in the conservative euler equation positive regardless of the direction of velocity?

The $$\rho u^2 + p$$ term corresponds to the flux of $$x$$-momentum through some surface. According to this equation it would give a positive momentum flux if $$u$$ was positive or negative. This does not make sense to me as if the velocity was reversed the momentum flux should be negative. Why is this term positive? ## 1 Answer

The equation is derived from momentum balance, so there is nothing wrong with. But you can resolve the apparent paradox like this. Take a square control volume $$C = [a,b] \times [c,d]$$ and look at x-momentum balance $$\frac{d}{dt}\int_C \rho u dx dy = -\int_c^d (\rho u^2)(b,y) dy + \int_c^d (\rho u^2)(a,y) dy + \ldots$$ The first term on the right is the convective flux of momentum across the right face of $$C$$. In first term, if $$u > 0$$, fluid is going out of $$C$$ and you lose x-momentum. On the other hand, if $$u < 0$$, fluid is entering $$C$$ but you still lose x-momentum because the incoming fluid has negative x-momentum. So in either case, you lose x-momentum due to momentum convection across the right face. The situation is reversed for the left face, you always gain x-momentum due to convective flux across the left face.

• Thanks that makes total sense! – Frosty May 3 at 5:38