Introduction to CFD and Fluid Mechanics Review

Computational Fluid Dynamics (CFD) is the numerical solution of the governing equations of fluid flow and heat transfer on a computer, using a discretised domain, to predict velocity, pressure, temperature and forces.

Brief history

• 1910s-1920s: Richardson and later Courant-Friedrichs-Lewy (1928) laid the ideas of finite-difference solution of PDEs and the stability condition.
• 1950s-60s: Digital computers allowed first numerical flow solutions (Los Alamos, Harlow and Welch's MAC and vortex methods); the stream-function-vorticity method was popular.
• 1970s: Panel methods, full-potential and Euler solvers for aerospace; the finite volume method (FVM) and the k-$\varepsilon$ model (Launder and Spalding) appeared.
• 1980s: SIMPLE-type pressure-velocity coupling (Patankar and Spalding) made incompressible solvers practical; commercial codes (PHOENICS, FLUENT, STAR-CD) were released.
• 1990s onwards: Unstructured meshes, RANS, LES, parallel computing, CAD-integrated tools; today CFD runs on workstations and clusters, often with GPU acceleration.

Role in mechanical engineering design

• Design of pumps, fans, turbines, compressors (blade loading, efficiency).
• IC engine in-cylinder flow, combustion and cooling passages.
• HVAC, cleanroom and electronics cooling, heat exchangers.
• Vehicle aerodynamics, drag reduction, wind loads on structures.
• Optimisation: many design variants are tested virtually before a prototype is built.

Advantages

• Lower cost and shorter design time than repeated prototypes.
• Full-field data (pressure, velocity, temperature everywhere), including places where probes cannot reach.
• Can simulate conditions that are dangerous or impossible to test (fire, very high temperature).
• Easy parametric study and optimisation.

Limitations

• Results depend on the model: turbulence, combustion and multiphase models are approximate.
• Numerical errors (truncation, round-off, poor mesh) and wrong boundary conditions give wrong answers.
• Needs skilled users, validation against experiments, and large computing time for 3D transient or LES/DNS cases.
• "Garbage in, garbage out": a coloured plot does not mean a correct answer.

A CFD analysis has three stages: pre-processing, solving and post-processing.

 Problem definition
        |
        v
+---------------------+
| 1. PRE-PROCESSING   |  geometry, mesh,
|                     |  physics, BCs
+---------------------+
        |
        v
+---------------------+
| 2. SOLVER           |<--+ not converged
|  discretise, iterate|   |
+---------------------+---+
        | converged
        v
+---------------------+
| 3. POST-PROCESSING  |  plots, forces,
|                     |  validation
+---------------------+

1. Pre-processing

1. Define the problem and the aim (e.g. pressure drop, lift).
2. Create or import the geometry (CAD) and clean it; choose the computational domain size.
3. Generate the mesh (structured/unstructured) and refine near walls and gradients.
4. Select the physical models: laminar/turbulent, steady/transient, compressible/incompressible, heat transfer.
5. Define fluid properties and boundary and initial conditions (inlet velocity, outlet pressure, wall, symmetry).

2. Solver

1. The governing equations are integrated over control volumes (FVM) or approximated by differences (FDM) to give algebraic equations.
2. A pressure-velocity coupling method (e.g. SIMPLE) and linear solvers (Gauss-Seidel, multigrid) solve them iteratively.
3. Iteration continues until the residuals fall to a set tolerance (e.g. $10^{-4}$ to $10^{-6}$) and monitored quantities stop changing.

3. Post-processing

• Contour, vector and streamline plots, line plots, animations.
• Integrated results: forces, pressure drop, heat transfer rate, mass-flow balance.
• Check mesh independence and compare with experimental or analytical data (verification and validation).
• If results are unsatisfactory, return to the mesh or the model and repeat.

Definitions

• Streamline: a curve that is tangent to the velocity vector of every fluid particle on it at one instant. Its equation is $\dfrac{dx}{u}=\dfrac{dy}{v}=\dfrac{dz}{w}$ (time held constant).
• Pathline: the actual trajectory of one fluid particle over a period of time. It is found from $\dfrac{dx}{dt}=u,\ \dfrac{dy}{dt}=v,\ \dfrac{dz}{dt}=w$.
• Streakline: the curve joining the positions, at one instant, of all particles that have passed through a fixed point earlier (e.g. smoke from a chimney or dye injected at a point).

Comparison

When they coincide

In steady flow the velocity at every point is independent of time, so a particle follows the same path as the streamline and all particles from one point follow it too. Hence streamlines, pathlines and streaklines are identical. In unsteady flow they are different.

Streamlines cannot cross because the velocity at a point has only one direction; there is no flow across a streamline, so a bundle of streamlines forms a streamtube.

Given: $u=2x$, $v=-2y$, steady 2D flow.

(a) Incompressibility

$$\nabla\cdot\vec V=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=2+(-2)=0$$

Since the divergence is zero, the continuity equation for incompressible flow is satisfied.

(b) Streamline through (2, 1)

$$\frac{dx}{u}=\frac{dy}{v}\ \Rightarrow\ \frac{dx}{2x}=\frac{dy}{-2y}\ \Rightarrow\ \frac{dx}{x}+\frac{dy}{y}=0$$

Integrating gives $\ln x+\ln y=\ln C$, so $xy=C$.

At (2, 1): $C=2\times1=2$.

Streamline: $xy = 2$ (a rectangular hyperbola). This is a stagnation-point flow: fluid approaches the origin along the y-axis and leaves along the x-axis..

(c) Acceleration

The flow is steady, so only the convective part remains:

$$a_x=u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=(2x)(2)+0=4x$$ $$a_y=u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=0+(-2y)(-2)=4y$$

At (2, 1): $a_x=8\ \text{m/s}^2$, $a_y=4\ \text{m/s}^2$.

$$|a|=\sqrt{8^2+4^2}=8.94\ \text{m/s}^2$$

Direction: $\tan^{-1}(4/8)=26.6^\circ$ above the x-axis.

(d) Rotation

$$\omega_z=\frac12\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)=\frac12(0-0)=0$$

The flow is irrotational.

Answer: (a) $\nabla\cdot\vec V=0$; (b) $xy=2$; (c) $a=8.94\ \text{m/s}^2$ ($a_x=8$, $a_y=4$ m/s²); (d) irrotational.

Definition and meaning

The Reynolds number is the ratio of inertia force to viscous force:

$$Re=\frac{\rho V L}{\mu}=\frac{VL}{\nu}$$

For pipe flow $L=D$. A small $Re$ means viscous forces dominate and disturbances are damped (laminar flow); a large $Re$ means inertia dominates, disturbances grow and the flow becomes turbulent.

Typical limits for pipes: laminar for $Re<2300$, transitional for about 2300 to 4000, turbulent for $Re>4000$.

(a) Reynolds number

$$Re=\frac{998\times0.9\times0.04}{1.002\times10^{-3}}=3.59\times10^{4}$$

Since $Re\gg4000$, the flow is turbulent.

(b) Velocity for $Re=2300$

$$V=\frac{Re\,\mu}{\rho D}=\frac{2300\times1.002\times10^{-3}}{998\times0.04}=0.0577\ \text{m/s}$$

So only a very slow flow (5.8 cm/s) is laminar in this pipe.

(c) Importance in CFD

• It decides whether a laminar solver or a turbulence model is required.
• It sets the mesh needs: higher $Re$ gives thinner boundary layers, so finer wall meshes (small $y^+$) are needed.
• It is the similarity parameter: a CFD model at small scale with the same $Re$ represents the full-size case.
• Drag coefficient, friction factor and transition behaviour are all functions of $Re$.

Answer: $Re=3.59\times10^4$ (turbulent); $V_{Re=2300}=0.0577$ m/s.