Verifying a Classic: Simulating Poiseuille Flow with OpenFOAM and ParaView - the engineer in me is still alive...

---

🚀 Introduction

In computational fluid dynamics (CFD), verifying the solution of known analytical problems helps build confidence in simulation tools and numerical schemes. One such classical benchmark is Poiseuille flow — a pressure-driven, fully developed laminar flow through a channel.

In this post, I recreate and verify the parabolic velocity profile of Poiseuille flow using OpenFOAM, and visualize it using ParaView.

---

📚 What Is Poiseuille Flow?

Poiseuille flow describes steady, incompressible, laminar flow of a Newtonian fluid in a long, straight channel driven by a pressure gradient.

✅ Assumptions

• Incompressible and Newtonian fluid
• Laminar, steady-state flow
• Fully developed: no changes along the flow direction
• No-slip boundary condition at the walls

✅ Governing Equation

From the Navier–Stokes equations, the simplified form for x-direction is:

$$\mu \frac{d^2 u}{dy^2} = \frac{dp}{dx}$$

Solving gives:

$$u(y) = \frac{1}{2\mu} \left( -\frac{dp}{dx} \right) y(H - y)$$

This is a parabola, symmetric across the channel height, with zero velocity at the walls and maximum velocity at the center.

---

🛠️ CFD Simulation with OpenFOAM

To simulate this in OpenFOAM:

✅ Geometry

• A 2D rectangular channel:
  • Length: 1.0 m
  • Height: 0.1 m
  • Depth: 0.01 m (or use empty for 2D)

✅ Boundary Conditions

Inlet:   fixed pressure (1 unit)
Outlet:  fixed pressure (0 unit)
Walls:   no-slip (U = 0)
Internal Field: uniform (0 0 0)

✅ Solver

Use icoFoam (transient, laminar, incompressible)

---

📊 Visualization in ParaView

After running the simulation:

1. Create a .foam file to open in ParaView
2. Use "Plot Over Line" tool
3. Define a vertical line at mid-channel (x=0.5), from y=0.0 to y=0.1
4. Plot the U_X component of velocity

📈 Result

The resulting plot shows the expected parabolic profile:

💡 Key Learnings

• Poiseuille flow is an ideal case for verifying laminar solvers
• Boundary condition correctness is crucial
• Mesh resolution affects symmetry and smoothness
• ParaView enables direct comparison to theory

---

🧠 Beyond the Code: What Poiseuille Flow Teaches Us

At first glance, Poiseuille flow is just a parabolic curve. But under the surface, it reveals deeper truths.

🌀 Pressure as a Metaphor

In this simulation, the fluid moves not because it wants to — but because pressure compels it. It obeys boundaries, yet flows optimally. This is how life often works — pressure shapes motion, boundaries define form.

🧩 The Parabola of Nature

That arc you see isn’t just math. It’s how rivers curve, how air flows through lungs, how blood pulses through arteries. It’s nature optimizing flow — again and again.

🧭 The Simulation Reflects You

As a solver:

• You define conditions
• You impose structure
• You wait for the system to converge

And maybe, just maybe, in the symmetry of that parabola, you catch a glimpse of your own search for balance.

---

✅ Conclusion

Poiseuille flow is more than a simulation — it’s a philosophical checkpoint. You’re not just verifying code. You’re using math and computation to echo nature.

You didn’t just run a case.

You discovered order, where theory meets flow, and code meets clarity.

---

Happy simulating. 🌊

So...

Here I am...

Enjoy...