This chapter introduces foundational incompressible flow archetypes used to understand and predict fluid behavior before resorting to full CFD simulations. Dimensional analysis, order-of-magnitude reasoning, and simplified flow models are used to identify dominant physical mechanisms, guide modeling assumptions, and anticipate flow features such as boundary layers, separation, and irrotational regions. The chapter builds physical intuition that directly informs meshing, boundary condition selection, and solver setup in CFD.

Why Study Simplified Incompressible Flows

Even though CFD can solve complex flows numerically, engineering insight comes from simplification.

Simplified flow analysis allows engineers to:

Anticipate dominant forces and transport mechanisms
Identify relevant length and velocity scales
Reduce unnecessary model complexity
Detect unphysical CFD results early

Incompressible flow is used as a baseline because:

Density variations are negligible in many liquid and low-speed gas flows
The physics becomes clearer without compressibility effects

Dimensional Analysis as a Thinking Tool

Dimensional analysis is not about formulas; it’s about structure.

Its purpose is to:

Reduce the number of governing parameters
Identify similarity between seemingly different flows
Reveal which effects matter and which do not

In CFD practice, dimensional analysis helps:

Decide whether viscosity matters
Decide if a flow is laminar or turbulent
Estimate boundary layer thicknesses
Scale results between models and real systems

Most importantly:

Dimensional analysis guides modeling before meshing or solving begins.

Non-Dimensional Thinking

Flows are governed by ratios of physical effects, not absolute values.

Typical questions an engineer asks:

Is inertia stronger than viscosity?
Is unsteadiness relevant?
Do pressure gradients dominate?

These questions lead to non-dimensional parameters that:

Classify flow regimes
Indicate expected flow behavior
Define validity ranges for simplifications

CFD models implicitly assume certain parameter ranges. Recognizing this avoids misuse.

Unidirectional Flow: The Simplest Archetype

5.1 Physical Meaning

Unidirectional flow:

Velocity points predominantly in one direction
Variations occur mainly across the flow, not along it

Examples:

Flow between parallel plates
Fully developed pipe flow
Slow viscous channel flows

These flows illustrate:

Balance between pressure forces and viscosity
How boundary conditions shape velocity profiles

5.2 Why It Matters in CFD

Unidirectional flows are:

Reference solutions
Validation benchmarks
Building blocks for more complex flows

They also explain:

Why mesh refinement is needed normal to walls
Why wall boundary conditions are critical
Why numerical diffusion can distort velocity profiles

Low Reynolds Number and Unsteady Effects

At low Reynolds numbers:

Viscous effects dominate
Flow responds smoothly to forcing
Inertia plays a secondary role

When unsteadiness is added:

Time scales become as important as length scales
Transient response reveals how momentum diffuses through the flow

This regime is especially important for:

Microfluidics
Lubrication flows
Early transient stages in larger systems

CFD in this regime is usually stable but sensitive to time-step selection.

Almost Unidirectional Flow: Boundary Layers

7.1 Boundary Layer Concept

In high Reynolds number flows:

Viscous effects are confined near solid walls
The outer flow behaves nearly inviscid

This leads to the boundary layer:

A thin region adjacent to walls
Strong velocity gradients normal to the surface
Dominant viscous dissipation

7.2 Why Boundary Layers Control Everything

Despite being thin, boundary layers:

Determine wall shear stress
Control heat and mass transfer
Trigger flow separation
Strongly influence drag

In CFD terms:

Boundary layers dictate mesh requirements more than any other flow feature.

7.3 Boundary Layer Growth and Separation

As flow develops along a surface:

Boundary layers thicken
Momentum near the wall decreases

If an adverse pressure gradient is strong enough:

Near-wall flow can reverse
Separation occurs

Separation fundamentally changes:

Pressure distribution
Wake structure
Turbulence production

Predicting separation is one of the hardest CFD challenges.

Irrotational Incompressible Flow

8.1 Physical Interpretation

Irrotational flow:

Has negligible vorticity
Velocity derives from a scalar potential

This approximation applies when:

Viscosity is negligible
Flow is away from walls and wakes

Examples:

Outer flow around streamlined bodies
Flow far from solid boundaries

8.2 Why It Still Matters

Irrotational flow models:

Provide analytical insight
Explain pressure distributions
Serve as outer solutions matched to boundary layers

In CFD:

Inviscid solvers or potential flow reasoning help debug pressure fields
Large-scale flow direction is often captured even when viscous effects are simplified

Simplified Flow Analysis: Tank Discharge

The tank discharge problem illustrates engineering decomposition:

Break a complex problem into simpler subproblems
Apply different assumptions in different regions
Recombine results consistently

Key lessons:

Not all regions require the same model fidelity
Order-of-magnitude estimates guide modeling effort
Preliminary analysis can predict trends before CFD is run

This mindset directly transfers to CFD domain decomposition.

Connection to Meshing Workflows

The meshing workflows introduced in the application chapters matter because:

Boundary layers require structured near-wall resolution
Regions of expected separation need local refinement
Irrotational regions tolerate coarser meshes

Watertight vs fault-tolerant workflows differ in geometry handling, but:

The physics, not the workflow, determines where resolution is required.

Meshing strategy should reflect:

Expected flow archetypes
Dominant gradients
Sensitivity regions

Engineering Intuition

Simplified flows are not obsolete, but act as diagnostic tools
Dimensional analysis prevents over-modeling
Boundary layers dominate wall-bounded flows
Separation is a global phenomenon triggered locally
Inviscid reasoning still explains much of the pressure field

A good CFD engineer always asks:

“Which simplified flow does this region resemble?”

Study Priorities

If short on time, focus on:

Purpose of dimensional analysis
Unidirectional vs almost unidirectional flow
Boundary layer concept and importance
Physical origin of separation
Role of irrotational flow models
Problem decomposition strategy

Key Takeaways

Incompressible flow analysis builds physical intuition before CFD.
Dimensional analysis reduces complexity and guides assumptions.
Boundary layers control shear, transfer, and separation.
Simplified models explain dominant mechanisms.
CFD results must be interpreted through these physical lenses.

Introduction to CFD Chapter 3: Basic Incompressible Flow Analysis