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

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?"

• The Couette and Poiseuille flow cases are pretty standard and pretty easy to reproduce, but they're not very interesting. Manufacturing 2D and 3D incompressible flows is pretty easy, though, so what's your goal? – Bill Barth Apr 17 '13 at 13:08
• Can you be more specific about what you want to test? Are you looking for a system with an analytic solution to which you can compare your numerics? – Ben Apr 17 '13 at 22:14
• @JohnDelong Yes I want to compare results of my 1-D algorithm with analytic or any well established test case (like 2-D lid driven cavity solution by Ghia et al). – Shainath Apr 18 '13 at 3:38

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.

• Thanks for writing this out. It's a strong contrast from what's possible in compressible flows. +2 – Bill Barth Apr 18 '13 at 16:32
• Perhaps you could make it less boring by adding a source term in the continuity equation, like in cavitation models. Keeping a divergence-free velocity field also limits the choice of manufactured solutions in 2D or 3D. Regarding the flat plate, there's also "triple deck theory" for a finite plate with a wake. You can go up to Re=$10^5$ and still be laminar. – chris Apr 19 '13 at 6:39
• @chris: There is virtually no limitation for manufactured solutions in 2 and 3 dimensions. You can always use a streamfunction in 2D and a vector streamfunction or cross product of gradients in 3D. – Bill Barth Apr 19 '13 at 17:59
• @BillBarth: that sounds interesting, I'll post a separate question on this subject. – chris Apr 22 '13 at 7:50

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)

• Yes,I am aware of these 2-D test cases for incompressible flow codes. I am search of 1-D case because, 1-D problems are beneficial if you want to try different algorithms on it within no time, and are easy to code. May be type of equation of incompressible flow (elliptic-parabolic) is restriction over there.Thanks for sharing your views. – Shainath Apr 18 '13 at 12:08
• Beyond the quasi 1-D cases already given, there are no additional 1-D cases of interest due to the incompressibility condition. In 1-D, it says $\frac{\partial{u}}{\partial{x}}=0$ which is pretty boring. – Bill Barth Apr 18 '13 at 12:20