Couette flow

Category: Fluid Analysis (CFD) | Integrated 2026-04-06

Couette Flow

Couette flow: Theoretical Foundations

Overview

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Professor, what is Couette flow?

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It's a shear flow of a viscous fluid between two parallel plates, driven by one plate moving at velocity $U$. Since an exact solution exists, it's one of the most fundamental problems used for verifying the accuracy of CFD codes.

Exact Solution

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What does the exact solution look like?

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For fully developed plane Couette flow, the velocity profile becomes a simple linear distribution.

$$ u(y) = U \frac{y}{h} $$

Here, $h$ is the distance between plates, and $y$ is the distance from the stationary plate.

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The shear stress is uniform regardless of distance from the wall,

$$ \tau = \mu \frac{U}{h} = \mu \dot{\gamma} $$

where $\dot{\gamma} = U/h$ is the shear rate.

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So the velocity just changes linearly. That's simple.

Couette-Poiseuille Flow

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When a pressure gradient exists in addition to plate movement, it becomes Couette-Poiseuille flow.

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

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The first term is the Couette component (linear), and the second term is the Poiseuille component (parabolic). Depending on the sign and magnitude of the pressure gradient, diverse velocity profiles such as favorable pressure gradient, adverse pressure gradient, and reverse flow can occur.

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So it's a superposition of Couette and Poiseuille flows.

Taylor-Couette Flow

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Couette flow between coaxial double cylinders is called Taylor-Couette flow, driven by the rotation of the inner cylinder. When the Taylor number

$$ Ta = \frac{\Omega^2 R_i (R_o - R_i)^3}{\nu^2} $$

exceeds the critical value $Ta_c \approx 1708$, Taylor vortices appear. This is a bifurcation phenomenon due to centrifugal instability and is a classic problem in pattern formation.

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Taylor vortices are those beautiful ring-shaped patterns, right?

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Yes. As Re increases further, it transitions to wavy vortex, modulated wavy vortex, and turbulent Taylor vortex. This route is a textbook case of turbulent transition.

Coffee Break Trivia

Why Couette Flow Supports Food Factories

Couette-type rheometers—devices that sandwich a sample between two plates and rotate one—are standard equipment for viscosity measurement in the food and cosmetics industries. Mayonnaise, chocolate, shampoo, toothpaste... these are non-Newtonian fluids whose viscosity changes with shear rate, so they cannot be measured with simple tube viscometers. By assuming the theoretical solution of Couette flow (linear velocity distribution), viscosity under the same shear rate conditions can be accurately derived. In the quality control departments of food manufacturers, checking product viscosity with a Couette-type rheometer every morning is a daily routine. The seemingly mundane theory supports our daily dining tables.

Computational Methods for Couette flow

Numerical Methods

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How do I solve Couette flow numerically?

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Plane Couette flow reduces to a 1D problem, so it's very simple.

Discretization of Plane Couette Flow

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Under fully developed conditions, $\partial u/\partial x = 0$, so the governing equation becomes

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

For pure Couette flow ($dp/dx = 0$), then

$$ \frac{d^2 u}{dy^2} = 0 $$

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Discretizing this with central differences gives

$$ \frac{u_{j+1} - 2u_j + u_{j-1}}{\Delta y^2} = 0 $$

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So it can be solved with a tridiagonal matrix.

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Yes. Boundary conditions are $u(0) = 0$ (stationary plate), $u(h) = U$ (moving plate). Even with second-order central differences, it matches the exact solution regardless of cell count. That's why it's used for verification.

Numerical Methods for Taylor-Couette Flow

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Taylor-Couette flow is axisymmetric, but the emergence of Taylor vortices is a 3D phenomenon, so all three components $(r, \theta, z)$ are necessary.

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Choice of numerical method:

Spectral Method: Fourier in circumferential direction, Fourier in axial direction (periodic BC), Chebyshev in radial direction
Finite Volume Method: Structured grid in cylindrical coordinates. Refine near walls in $r$ direction

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What resolution is needed to capture Taylor vortices with the finite volume method?

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A minimum of 20-30 cells in the radial direction and 10-15 cells per Taylor vortex wavelength in the axial direction is a guideline. The axial length of the computational domain should be an integer multiple of the vortex wavelength (typically 4-8 wavelengths) and use periodic boundary conditions.

Comparison with Stability Analysis

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To determine the critical Ta number for Taylor vortices via numerical computation, start from an initial condition with a small perturbation and observe the growth/decay of the perturbation over time. If it matches the theoretical $Ta_c = 1708$ within 2%, the code's validity is confirmed.

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Reproducing the critical stability value numerically is a good test of accuracy.

Coffee Break Trivia

What Taylor Vortices Teach Us: "The Harbinger of Numerical Instability"

In numerical calculations of Couette flow, when you increase the rotation speed, at some point the velocity field may develop periodic striped patterns. This is a phenomenon called "Taylor vortices," a structural instability that occurs when centrifugal force exceeds viscous force. When vortices suddenly appear in the calculation, it's easy to think, "Is this a bug?" but it's actually the physically correct solution. If you calculate the Taylor number $Ta$, the transition should occur at the theoretical critical value. In other words, you can use Couette flow to benchmark "whether your code can correctly capture vortex generation." It is also used to verify the operation of turbulence models.

Couette flow in Practice

Practical Guide

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Please teach me the specific method for using Couette flow for CFD verification.

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Let's proceed step by step.

Step 1: Plane Couette Flow (Code Verification)

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Purpose: Confirm the order of accuracy of spatial discretization

Setup: Between two flat plates ($h = 1$ m), top plate $U = 1$ m/s, $\nu = 0.01$ m$^2$/s ($Re = 100$)

Mesh: Four levels: $N_y = 5, 10, 20, 40$

Check: Whether the $L_2$ error of the velocity profile decreases with mesh refinement as $O(\Delta y^p)$ to the power $p$

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