Sources and References

• Versteeg and Malalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, 2nd ed., Pearson.
• Ferziger, Perić, and Street, Computational Methods for Fluid Dynamics, 4th ed., Springer.
• Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere.
• Wilcox, Turbulence Modeling for CFD, 3rd ed., DCW.
• Pope, Turbulent Flows, Cambridge University Press.

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Chapter 1: Governing Equations and Discretization

Computational fluid dynamics converts the partial differential equations of fluid motion into algebraic systems solvable by computer. The designer who uses CFD as a tool must understand both the equations being solved and the numerics used to solve them, because modelling choices — turbulence closure, boundary-condition treatment, mesh resolution — shape results far more than the solver’s brand name.

1.1 Conservation Equations

Mass, momentum, and energy conservation for a Newtonian fluid are

\[ \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho \mathbf{u}) = 0, \]\[ \frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla\cdot(\rho \mathbf{u}\otimes\mathbf{u}) = -\nabla p + \nabla\cdot\boldsymbol{\tau} + \rho \mathbf{g}, \]\[ \frac{\partial (\rho e_0)}{\partial t} + \nabla\cdot(\rho e_0 \mathbf{u}) = -\nabla\cdot(p \mathbf{u}) + \nabla\cdot(\boldsymbol{\tau}\cdot\mathbf{u}) - \nabla\cdot\mathbf{q} + \dot{Q}, \]

with viscous stress tensor \( \boldsymbol{\tau} = \mu(\nabla\mathbf{u} + \nabla\mathbf{u}^T) - \frac{2}{3}\mu(\nabla\cdot\mathbf{u})\mathbf{I} \) and heat flux \( \mathbf{q} = -k\nabla T \). Incompressible flow simplifies to constant \( \rho \) and decouples energy from momentum unless buoyancy matters.

1.2 Finite-Volume Discretization

The finite-volume method integrates conservation laws over control volumes, converting volume integrals of divergence to surface integrals of flux. For a generic transport equation

\[ \frac{\partial (\rho \phi)}{\partial t} + \nabla\cdot(\rho \mathbf{u} \phi) = \nabla\cdot(\Gamma \nabla \phi) + S, \]

integration over cell \( P \) gives

\[ \frac{d(\rho \phi V)_P}{dt} + \sum_f (\rho \mathbf{u}\cdot\mathbf{A}\phi)_f = \sum_f (\Gamma \nabla\phi\cdot\mathbf{A})_f + S_P V_P. \]

Convective flux requires interpolation of \( \phi \) to faces. Upwind schemes are stable but introduce numerical diffusion; central schemes are accurate but produce oscillations at high Péclet number. Modern practice uses second-order upwind or higher-order flux-limiter schemes that balance accuracy and boundedness.

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Chapter 2: Pressure–Velocity Coupling

2.1 The Pressure Problem

In incompressible flow, pressure acts as a Lagrange multiplier enforcing continuity; it has no independent evolution equation. Staggered grids, collocated grids with Rhie–Chow interpolation, or projection methods decouple velocity and pressure.

2.2 Segregated Solvers

SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) solves momentum with a guessed pressure, computes a pressure correction from the continuity equation, corrects velocity and pressure, and iterates. Variants SIMPLER, SIMPLEC, and PISO address specific weaknesses: PISO adds a second corrector stage for transient accuracy.

The pressure-correction equation is a Poisson-like elliptic system whose solution dominates solver cost. Algebraic multigrid (AMG) accelerates it by coarsening the problem and relaxing on a hierarchy of grids, producing near-optimal convergence rates for elliptic operators.

2.3 Coupled and Compressible Solvers

Coupled solvers treat momentum and continuity as a single large system. They converge in fewer iterations but each iteration costs more; on large meshes the trade-off often favours segregated methods. Compressible solvers use density-based schemes with flux splitting and shock-capturing; preconditioning extends them to low-Mach regimes.

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Chapter 3: Iterative Solution of Sparse Linear Systems

3.1 Matrix Structure

Finite-volume discretization produces sparse matrices with structured or unstructured sparsity patterns. Diagonal dominance is required for many iterative schemes; schemes such as deferred correction maintain it even with higher-order fluxes.

3.2 Stationary Iterations

Jacobi, Gauss–Seidel, and successive over-relaxation (SOR) are stationary iterative methods. Their convergence rate per iteration is poor for large problems; they serve as smoothers within multigrid.

3.3 Krylov Methods

Conjugate gradient (for symmetric positive-definite systems), BiCGStab, and GMRES are Krylov-subspace methods. Preconditioning with incomplete LU or with multigrid is essential for good performance.

3.4 Multigrid Acceleration

Geometric multigrid uses a sequence of grids \( h, 2h, 4h, \ldots \) and couples smoothing on each grid with restriction and prolongation operators. Algebraic multigrid constructs the coarsening from the matrix graph, allowing application to unstructured meshes. V-, W-, and F-cycles define the traversal pattern; typical CFD practice pairs AMG with BiCGStab to handle anisotropic meshes and mildly indefinite operators.

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Chapter 4: Turbulence Modelling

4.1 Reynolds Averaging

Decomposing velocity and pressure into mean and fluctuating parts \( \mathbf{u} = \bar{\mathbf{u}} + \mathbf{u}' \) and substituting into momentum yields the Reynolds-averaged Navier–Stokes (RANS) equations with the Reynolds stress tensor

\[ \tau^R_{ij} = -\rho \overline{u_i' u_j'}, \]

which must be modelled to close the equations. Eddy-viscosity models invoke the Boussinesq hypothesis,

\[ \tau^R_{ij} = \mu_t\!\left(\frac{\partial \bar{u}_i}{\partial x_j} + \frac{\partial \bar{u}_j}{\partial x_i}\right) - \frac{2}{3}\rho k \delta_{ij}, \]

relating Reynolds stress to mean strain through a scalar turbulent viscosity.

4.2 Two-Equation Models

The k–ε model solves transport equations for turbulent kinetic energy \( k \) and dissipation rate \( \varepsilon \) and computes \( \mu_t = C_\mu \rho k^2/\varepsilon \). Standard k–ε handles free shear flows well but struggles near walls and under adverse pressure gradient. The k–ω model replaces \( \varepsilon \) with specific dissipation \( \omega \) and behaves better at walls. Menter’s SST model blends k–ω near walls with k–ε in the freestream, capturing separation more reliably.

4.3 Reynolds Stress Models and LES

Reynolds stress models solve transport equations for each component of \( \tau^R \). They capture anisotropy — critical in swirling flows, rotating machinery, and flows with strong streamline curvature — but are more expensive and temperamental. Large-eddy simulation (LES) resolves the large, energy-containing turbulence structures and models only the subgrid scales; it is dramatically more expensive than RANS and requires fine meshes, but it predicts unsteady phenomena (vortex shedding, combustion instability, acoustic noise) that RANS cannot.

4.4 Wall Treatment

Near-wall turbulence has a universal structure (viscous sublayer, buffer, log-law). Wall functions map the first cell to the log law and tolerate relatively coarse near-wall meshes (\( y^+ > 30 \)). Low-Reynolds wall-integration resolves the viscous sublayer directly and requires \( y^+ < 1 \), with ten or more cells within \( y^+ = 30 \). The choice is governed by the flow physics the simulation must capture: wall heat transfer and separation require integration to the wall.

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Chapter 5: Boundary Conditions, Meshing, and Practice

5.1 Boundary Conditions

Inlet conditions specify velocity, pressure, temperature, and turbulence quantities; outlet conditions typically impose pressure or extrapolate. Wall conditions depend on the turbulence treatment. Symmetry, periodic, and cyclic conditions reduce mesh size when the geometry permits. Radiation, species, and multiphase boundary conditions extend the framework for specific problems.

5.2 Mesh Generation

Meshes are structured (hexahedral, ordered) or unstructured (tetrahedral, polyhedral, hybrid). Hex meshes are accurate and efficient per cell but expensive to generate on complex geometry. Unstructured meshes automate generation on arbitrary geometry. Polyhedral meshes, now widely available, offer good accuracy and convergence with far fewer cells than tetrahedral equivalents. Boundary-layer meshes extrude thin layers of cells from walls to resolve near-wall gradients.

Mesh independence is established by systematic refinement and extraction of an asymptotic solution; the Grid Convergence Index (GCI) quantifies discretization uncertainty.

5.3 Verification and Validation

Verification confirms that the solver correctly solves the equations (mesh convergence, code verification against manufactured solutions). Validation compares the simulation to experiment and quantifies predictive accuracy. Design use of CFD demands both: the ASME V&V 20 standard provides a formal framework.

5.4 Applied Design Studies

Classroom applications include flow in a sudden expansion (validation against Armaly’s benchmark), backward-facing step (separation and reattachment prediction), and a centrifugal-compressor stage (swirl, turbomachinery interfaces, mixing-plane treatment). Each study exposes a specific modelling choice and its consequence for the engineering prediction of interest.

Effective CFD practice is iterative: set up the simplest model that could answer the question, validate it, then add complexity only where justified by discrepancy from experiment or by a need for finer detail in the design decision being supported.