Pressure-Pressure BC

Question

My FDM code simulates Backward Facing Step flow when I use conventional BCs such as defining velocity profile at inlet and fully developed condition at outlet. I have validated the results and it seems functioning correctly.

However, when I want to impose a pressure-pressure condition (both static), it only converges when a zero-gradient condition is selected for inlet. $$ @inlet: p=p_{in} \hspace{10 mm} ,\dfrac{du}{dx}=0 \hspace{10 mm}, \dfrac{dv}{dx}=0 $$ $$ @outlet: p=p_{out} \hspace{10 mm},\dfrac{du}{dx}=0 \hspace{10 mm}, \dfrac{dv}{dx}=0 $$ When I try to use mass balance to update inlet normal velocity at each iteration, the solution eventually blows up. I am integrating outflow and use the mean velocity as a uniform inlet velocity. $$ @inlet: p=p_{in} \hspace{10 mm} ,u=U_{in} \hspace{10 mm}, v=0 $$ $$ @outlet: p=p_{out} \hspace{10 mm},\dfrac{du}{dx}=0 \hspace{10 mm}, \dfrac{dv}{dx}=0 $$ where $U_{in}$ is found from last iteration mass balance.

Also I am interested to know if there is any difference to use momentum balance and mass balance to update inlet normal velocity.

Has anyone faced this problem before? I appreciate your help

Answer 1

A complication arises when you specify both the mass balance and the pressure differnce for an incompressible fluid because the pressure and velocity are coupled. A certain pressure difference can only have a particular set of velocities in the domain which satisfy the Navier Stokes equations. At the grass root level it can be explained by applying the Bernoulli's equation for a one dimensional problem at the INLET and the OUTLET

\begin{equation*} p_{in}+\dfrac{1}{2}\rho V_{in}^2 = p_{out}+\dfrac{1}{2}\rho V_{out}^2 \end{equation*}

If you were specifying the mass balance $V_{in}$ and pressure differnce $p_{out}-p_{in}$ you would be only left with $V_{out}$ in an incompressible fluid, which sounds reasonable. However the principle of mass conservation also has to be religioulsy followed which is why the solver fails. The mass balance equation is written as:

\begin{equation} A_{in}V_{in}=A_{out}V_{out} \end{equation} where $A_{in}$ and $A_{out}$ are the cross-sectional areas of the INLET and the OUTLET. It would be satisfied only if the INLET velocity $V_{in}$ prescribed by you and the OUTLET velocity $V_{out}$ obtained through the Bernouli equation matches with the ratios of the cross sectional areas as well. However this is rarely the case, and thus the equations you solve in the numerical solver, fail the check for conservation of mass. This causes the solver to explode.

Do note that this would not be the case if the solver was compressible. The density $\rho$ would adjust itself accordingly to satisfy the Navier Stokes equations and also the conservation of mass.

To asnwer your second question, if specifying the mass balance is the same as applying the momentum balance. Again it boils down to the compressibility of the fluid. If the fluid is compressible then the mass balance is not the same as applying the momentum balance. However if you are using an incompressible fluid, the momentum inlet would simply be $\rho $ times the velocity inlet.