Is there any 1-D test case for incompressible flow codes?

Question

There is quasi 1-D C-D nozzle test case for compressible flow codes, like that "is there any 1-D test case for incompressible flow codes?"

Answer 1

To build on Bill Barth's commment, there really isn't any depth here. The equations you're trying to solve are: \begin{align} u_t + uu_x &= - \frac{1}{\rho}p_x + \nu u_{xx} \\ u_x & = 0 \end{align} Now, $u_x = 0$ immediately implies $u$ has no space-dependence, so we can write $u = f(t)$, for some time-dependent function $f: \mathbb{R} \to \mathbb{R}$. The momentum equation then reduces to $$ f'(t) = -\frac{1}{\rho}p_x(x,t).$$ Integrating this, we get $$p = -\rho x f'(t) + g(t),$$ where $g :\mathbb{R} \to \mathbb{R}$ is another function of time. (Note that $g$ is irrelevant to the dynamics of the system, as there is dependence only on the spatial derivatives of $p$.)

At this point, the solution space is small and uninteresting, and we haven't even applied boundary conditions. Homogenous Dirichlet boundary conditions on $u$, for example, imply that $u \equiv 0$ is the only solution. Even periodic boundary conditions, by requiring that $p$ be periodic, lead to the condition that $f'(t) \equiv 0$, so $u \equiv C$ for some constant $C$.

In short, what you're looking for simply doesn't exist.

Answer 2

Assuming that the system you are interested in is not hyperbolic, (shallow water equations model flow of an incompressible fluid and yet are hyperbolic), Following could be taken as test cases for validation of the code. (although none of them is purely 1d)

1) As per Bill Barth's suggestion, Couette and Poiseuille flow problems are standard test cases

2) Uniform flow over the flat plate, developing laminar boundary layer can be solved. Its behavior is given by the well-established Blasius solution.

(The reason of why there are not a plenty 1d problems available, might be as follows:

In hyperbolic systems, 1d test / benchmark problems are plenty. The reason being that wave formation and transmission are more dominant phenomena in hyperbolic systems. Shocks and contacts can be analysed as a 1d phenomena (In certain 2d problems, these can be converted into 1d and then solved). But in incompressible flow, no purely 1d phenomenon takes place)