This chapter introduces the finite volume method (FVM), the numerical framework underlying most industrial CFD solvers. It explains how conservation laws are enforced on discrete control volumes, how fluxes are approximated at cell faces, how pressure–velocity coupling is achieved for incompressible flows, and how solver stability, convergence, and post-processing should be interpreted from an engineering perspective

Why the Finite Volume Method Dominates CFD

The finite volume method is built directly on integral conservation laws.

This matters because:

Conservation of mass, momentum, and energy is enforced by construction
Discontinuous solutions (e.g. strong gradients, shocks in compressible flow) are naturally admissible
Complex geometries can be handled using arbitrary polyhedral cells

Compared to finite differences or finite elements:

FVM prioritizes physical conservation over mathematical elegance
This makes it robust for industrial flows involving turbulence, multiphase physics, and strong nonlinearities

Control Volumes and Discrete Thinking

In FVM, the domain is decomposed into finite control volumes (cells).

Each cell:

Represents a small but finite portion of the flow
Stores averaged values of flow variables
Exchanges fluxes with neighboring cells through faces

Key idea:

CFD does not solve equations at points, but balances fluxes across surfaces.

This surface-based interpretation explains why:

Face normals and areas matter
Mesh quality directly affects accuracy
Conservation errors are localized, not global

Collocated vs Non-Collocated Variable Storage

4.1 What This Means Physically

Collocated schemes: all variables stored at cell centers
Non-collocated (staggered) schemes: velocities stored at faces, scalars at centers

Trade-off:

Non-collocated schemes reduce pressure–velocity decoupling
Collocated schemes are more flexible for complex, unstructured meshes

Modern CFD (including Fluent) favors collocated schemes, but compensates using:

Pressure correction techniques
Interpolation and stabilization strategies

4.2 Engineering Implication

Pressure oscillations or checkerboarding are numerical artifacts, not physical effects.
When they appear, they indicate:

Insufficient coupling
Poor mesh quality
Inappropriate solver settings

Fluxes: How Physics Crosses Cell Faces

Every transport equation can be understood as:

Convective transport (motion with the flow)
Diffusive transport (smoothing due to gradients)
Sources (generation or removal)

In FVM:

Fluxes are evaluated at faces
Cell-center values must be interpolated to faces

The choice of interpolation controls:

Accuracy
Stability
Numerical diffusion

Convective Discretization: Accuracy vs Robustness

Convective schemes range from:

Low-order (upwind): stable, diffusive
High-order (QUICK, MUSCL): accurate, less dissipative

Engineering interpretation:

First-order schemes smear gradients but rarely diverge
Higher-order schemes capture sharp features but amplify instability

Rule of thumb:

Start robust, then increase accuracy once the flow is stable.

This is why many industrial workflows begin with first-order and later switch to higher-order discretization

Temporal Discretization and Pseudo-Time

Even steady problems are solved iteratively in pseudo-time.

Key distinction:

Physical time step → represents real transient evolution
Pseudo-time step → numerical device to reach steady state

Stability depends on:

Courant number
Mesh size
Flow velocity

High Courant numbers:

Accelerate convergence
Increase risk of divergence

Low Courant numbers:

Improve stability
Increase computational cost

Pressure-Based Solvers for Incompressible Flow

8.1 Why Pressure Needs Special Treatment

For incompressible flow:

Density is constant
Pressure does not have its own evolution equation

Instead:

Pressure emerges as a constraint enforcing mass conservation

This leads to pressure–velocity coupling algorithms.

8.2 SIMPLE, SIMPLEC, and PISO (Conceptual)

All pressure-based algorithms:

Guess a pressure field
Solve momentum equations
Correct pressure and velocity to enforce continuity

Differences lie in:

How aggressively corrections are applied
Whether the method targets steady or transient flows

Engineering guidance:

SIMPLE: robust default
SIMPLEC: faster convergence for simple flows
PISO: preferred for transient problems or large time steps

Under-Relaxation: Artificial Damping for Stability

Under-relaxation:

Limits how much variables change per iteration
Prevents runaway divergence

Important insight:

Under-relaxation affects convergence speed, not the final solution.

If residuals explode:

Reduce relaxation
Improve mesh
Revisit boundary conditions

If convergence is slow but stable:

Relaxation can be increased cautiously

This is numerical stabilization, not physics modeling.

Boundary Conditions as Part of the Numerical System

Boundary conditions are not “inputs”, they close the algebraic system.

Poorly posed boundaries cause:

Non-physical pressure fields
Artificial recirculation
Solver instability

Key ideas:

Inlet conditions define momentum and scalars
Outlet conditions must allow information to leave the domain
Over-constraining a boundary leads to numerical conflict

A CFD solution is only as good as its boundary condition logic.

Cell Zones and Material Modeling

Cell zones define:

Whether a region is fluid or solid
Which equations are solved
Which material properties apply

Engineering relevance:

Solid zones enable conjugate heat transfer
Porous zones introduce modeled momentum loss
Incorrect zoning silently corrupts results

Always verify:

Which equations are active
Which properties are temperature- or pressure-dependent

Post-Processing: Extracting Engineering Meaning

CFD produces massive datasets: post-processing is where engineering happens.

Useful outputs are rarely raw fields:

Forces, moments
Pressure drops
Mass and energy balances
Integral quantities over surfaces or volumes

Residual convergence alone is insufficient:

Always check physical balances
Compare inflow vs outflow
Verify expected trends

Visualization supports understanding but does not replace validation.

Engineering Intuition

FVM enforces conservation locally, face by face
Mesh quality is part of the numerical model
Pressure is a constraint, not a transported quantity
Stability tools do not change physics, just manage iteration
Post-processing determines whether CFD answers the engineering question

A good CFD engineer understands why the solver converges, not just that it does.

Study Priorities

If short on time:

Control volume interpretation
Convective vs diffusive flux meaning
Pressure–velocity coupling logic
Role of under-relaxation and Courant number
Boundary conditions as system closure
Difference between numerical convergence and physical correctness

Key Takeaways

Finite volume methods enforce conservation by construction.
Fluxes across faces are the core numerical mechanism.
Solver stability is managed through relaxation and time stepping.
Pressure-based solvers enforce incompressibility indirectly.
Engineering insight is extracted during post-processing, not during solving.

Introduction to CFD Chapter 4: Fundamentals of the Finite Volume Method