Computational Fluid Dynamics (CFD)

Introduction to CFD

• Computational Fluid Dynamics (CFD) predicts fluid flow, heat transfer, mass transfer, chemical reactions via numerical solutions of mathematical equations.

What CFD Does

• CFD simulates new designs, product development, troubleshooting, and redesign.
• CFD analysis outputs velocity, pressure, temperature, concentration, species, and forces distributions.

How CFD Works

• Pre-processing: Sets up the problem including geometry, mesh generation, physics, fluid properties, and boundary conditions.
• Solving: Equations are solved iteratively to reach a converged solution.
• Post-processing: Analyzes results and generates reports.

CFD Applications

• Aerospace: Aircraft aerodynamics, drag, and lift
• Automotive: Drag reduction, engine cooling
• Biomedical: Airflow in lungs, blood flow in arteries
• Chemical processing: Mixing, separation
• HVAC: Heating, ventilation, air conditioning
• Hydrology: Open channel flow, sediment transport
• Marine: Loads on offshore structures
• Nuclear: Thermal-hydraulics of reactors
• Power generation: Combustion, heat transfer

Conservation Laws

• Mass Conservation Equation: $\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot \left( {\rho V} \right) = 0$, where $\rho$ is density and $V$ is velocity vector.
• Momentum Conservation Equation: $\frac{{\partial \left( {\rho V} \right)}}{{\partial t}} + \nabla \cdot \left( {\rho VV} \right) = - \nabla p + \nabla \cdot \tau + \rho g + F$, where $p$ is pressure, $\tau$ is stress tensor, $g$ is gravitational acceleration, and $F$ is external force.
• Energy Conservation Equation: $\frac{{\partial \left( {\rho h} \right)}}{{\partial t}} + \nabla \cdot \left( {\rho Vh} \right) = \nabla \cdot \left( {k\nabla T} \right) + S_h$, where $h$ is enthalpy, $k$ is thermal conductivity, $T$ is temperature, and $S_h$ is the heat source.

Turbulence Modeling

• Turbulence is a chaotic fluid flow with random changes in time and space.
• Turbulent flow features 3D vorticity, fluctuations, dissipation, and diffusivity.
• Most engineering flows are turbulent.
• Direct Numerical Simulation (DNS) solves Navier-Stokes equations without modeling.
• Reynolds Averaged Navier-Stokes (RANS) solves Navier-Stokes equations with turbulence models.
• Large Eddy Simulation (LES) solves filtered Navier-Stokes equations; only large eddies are resolved.
• Detached Eddy Simulation (DES) is a hybrid RANS/LES model.

Numerical Methods

• Discretization:
  • Finite Difference Method (FDM)
  • Finite Volume Method (FVM)
  • Finite Element Method (FEM)
• Solution Algorithms:
  • Iterative methods
  • Explicit / Implicit schemes
  • Coupled / Segregated algorithms

CFD - Advantages

• Reduced lead times and costs
• Ability to study systems under hazardous conditions
• Practically unlimited level of detail of results
• Comprehensive information on all relevant parameters

CFD - Disadvantages

• Approximations to real physics are inevitable
• Accuracy depends on user skill
• Results have a degree of uncertainty
• CFD is a tool, not a replacement for physical experiments

Fourier Analysis

• It decomposes a function into simpler sinusoidal functions.
• It is useful for analyzing signals and solving differential equations.

Fourier Series

• It represents a periodic function as a sum of sines and cosines.
• The Fourier series of a function $f(x)$ with period $2L$ is defined as $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \left( \frac{n \pi x}{L} \right) + b_n \sin \left( \frac{n \pi x}{L} \right) \right)$.
• $\text{Where} \ a_0 = \frac{1}{L} \int_{-L}^{L} f(x) , dx$, $a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos \left( \frac{n \pi x}{L} \right) , dx$, and $b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin \left( \frac{n \pi x}{L} \right) , dx$

Convergence of the Fourier Series

• If $f(x)$ and $f'(x)$ are piecewise continuous on $[-L, L]$, the Fourier series converges to $f(x)$ if $f$ is continuous at $x$.
• If not, it converges to the average $\frac{f(x^+) + f(x^-)}{2}$ when $f$ is discontinuous at $x$.

Even and Odd Functions

• If $f$ is even, then $b_n = 0$ for all $n$.
• If $f$ is odd, then $a_n = 0$ for all $n$.

Examples of Fourier Series

• Step Function: If $f(x) = \begin{cases} 0, & -L < x < 0 \ E, & 0 < x < L \end{cases}$, then $f(x) = \frac{E}{2} + \frac{2E}{\pi} \sum_{n=1}^{\infty} \frac{\sin((2n-1)\pi x/L)}{2n-1}$.
• Linear Periodic Function: If $f(x) = x, \quad -L < x < L$, then $f(x) = \frac{2L}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin\left(\frac{n \pi x}{L}\right)$.

Fourier Transform

• The Fourier Transform is an extension of Fourier series to non-periodic functions.
• It transforms time-domain function to the frequency domain.

Definition of Fourier Transform

• Fourier Transform is the representation in frequency of $f(t)$, $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j \omega t} , dt \newline$ where: $F(\omega)$ is the representation in frequency of $f(t)$, $\omega$ is the angular frequency, and $j$ is the imaginary unit.

Inverse Fourier Transform

• To recover time-domain function $f(t)$ from frequency domain: $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j \omega t} , d\omega$

Properties of Fourier Transform

• Linearity: $\mathcal{F}{af(t) + bg(t)} = aF(\omega) + bG(\omega)$
• Scaling: $\mathcal{F}{f(at)} = \frac{1}{|a|}F\left(\frac{\omega}{a}\right)$
• Time Shifting: $\mathcal{F}{f(t - t_0)} = e^{-j \omega t_0} F(\omega)$
• Frequency Shifting: $\mathcal{F}{e^{j \omega_0 t} f(t)} = F(\omega - \omega_0)$
• Convolution in Time: $\mathcal{F}{(f * g)(t)} = F(\omega)G(\omega)$
• Convolution in Frequency: $\mathcal{F}{f(t)g(t)} = \frac{1}{2\pi}(F * G)(\omega)$
• Differentiation in Time: $\mathcal{F}\left{\frac{df}{dt}\right} = j\omega F(\omega)$
• Integration in Time: $\mathcal{F}\left{\int_{-\infty}^{t} f(\tau) , d\tau\right} = \frac{F(\omega)}{j\omega} + \pi F(0) \delta(\omega)$

Examples of Fourier Transforms

• Impulse Function (Dirac Delta): If $f(t) = \delta(t)$, then $F(\omega) = 1$.

• Constant Function: If $f(t) = 1$, then $F(\omega) = 2\pi \delta(\omega)$.

• Unit Step Function: If $f(t) = u(t) = \begin{cases} 0, & t < 0 \ 1, & t > 0 \end{cases}$, then $F(\omega) = \pi \delta(\omega) + \frac{1}{j\omega}$.

• Exponential Function: If $f(t) = e^{-at}u(t), \quad a > 0$, then $F(\omega) = \frac{1}{a + j\omega}$.

Applications of Fourier Analysis

• Applications include spectrum analyses of signals and designing filters
• Applications also feature data compression and differential resolution.

Chemical Kinetics

• $A + B \rightarrow C + D$
• Reaction Rate: the speed with which the concentration of a reactant decreases or the concentration of a product increases over time.
• Rate Law: Expresses the rate as a function of concentrations and temperature.

Rate Law

• The rate law must be experimentally determined.

• The rate constant, $k$, is temperature dependent.

• The exponents, $x$ and $y$, are the order of the reaction with respect to each reactant.

• If $rate = k[A]^x[B]^y$, the general form is $aA + bB \rightarrow cC + dD$ and $rate = -\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$.

Integrated Rate Laws

• The integrated rate law relates the concentration of a reactant to time.
• Zero Order:
  • Rate law: $rate = k$
  • Integrated rate law: $[A]_t = -kt + [A]_0$
  • Linear plot: $[A]_t$ vs. t
  • Slope: -k
  • y-intercept: $[A]_0$
  • Half-life: $t_{1/2} = [A]_0/2k$
• First Order:
  • Rate law: $rate = k[A]$
  • Integrated rate law: $ln[A]_t = -kt + ln[A]_0$
  • Linear plot: $ln[A]_t$ vs. t
  • Slope: -k
  • y-intercept: $ln[A]_0$
  • Half-life: $t_{1/2} = 0.693/k$
• Second Order:
  • Rate law: $rate = k[A]^2$
  • Integrated rate law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
  • Linear plot: $\frac{1}{[A]_t}$ vs. t
  • Slope: k
  • y-intercept: $\frac{1}{[A]_0}$
  • Half-life: $t_{1/2} = 1/k[A]_0$
• Second Order:
  • Rate law: $rate = k[A][B]$
  • Integrated rate law: $\frac{1}{[A]_0-[B]_0}ln(\frac{[B]_t[A]_0}{[A]_t[B]_0}) = kt$
  • Linear plot: $\frac{1}{[A]_t}$ vs. t
  • Slope: k
  • y-intercept: $\frac{1}{[A]_0}$

Collision Theory

• For a reaction to occur, reactant molecules must collide with:
  • Sufficient energy to break bonds.
  • Proper orientation.

Arrhenius Equation

• Relates the rate constant, $k$, to the activation energy, $E_a$, and temperature, $T$.
• $k = Ae^{-E_a/RT}$
• A is the frequency factor (related to the number of collisions that have the proper orientation).
• $ln(k) = ln(A) - \frac{E_a}{RT}$
• Taking the natural log of both sides of the Arrhenius equation gives a linear equation.
• $ln(\frac{k_1}{k_2}) = \frac{E_a}{R}(\frac{1}{T_2} - \frac{1}{T_1})$

Reaction Mechanisms

• A reaction mechanism is a series of elementary steps that must satisfy two requirements:
  • The sum of the elementary steps must give the overall balanced equation.
  • The mechanism must agree with the experimentally determined rate law.
• Intermediate: A species that is formed in one step and consumed in a subsequent step.
• Rate-determining step: The slowest step in the mechanism.
  • The rate law for the rate-determining step is the rate law for the overall reaction.
• Catalyst: A substance that speeds up a reaction without being consumed in the reaction.
  • A catalyst lowers the activation energy for the reaction.
  • A catalyst is consumed in one step and regenerated in a subsequent step.

Geometry and Algebra

• Geometry and algebra are related mathematics disciplines.
• Algebraic representation of geometric objects is a central concept.

Algebraic Representation of Geometric Objects

• Lines: in the Cartesian plane can be represented as $ax + by + c = 0$, where $a$, $b$, and $c$ are constants.
• Circles: with center $(h, k)$ and radius $r$ represented as $(x - h)^2 + (y - k)^2 = r^2$.
• Conics: like ellipses, hyperbolas, and parabolas, represented by second-degree algebraic equations.

Algebra and Geometric Problems

• Areas and volumes, formulas can be algebraically expressed.
• Geometric Theorems can be demonstrated using algebra.
• Geometric transformations like translations, rotations, and reflections can be represented with matrices and operations.

Analytic Geometry

• Analytic geometry focuses on studying geometry through algebraic methods.
• René Descartes introduced the Cartesian coordinate system.
• Points on a plane are represented as $(x, y)$.

Benefits of Analytic Geometry

• Improves over synthetic geometry: generality, precision, numerous applications.

Applications of Geometry and Algebra

• Manifest in:
  • computer graphics
  • CAD
  • GIS
  • robotics

Partial Differential Equations

• Chapter 2: The Heat Equation

Physical Model

• Derivation of the Heat Equation:

  • Assumptions:
    • Temperature depends on position $x$ and time $t$.
    • The rod is laterally insulated.
    • The rod is homogeneous with constant density $\rho$ and specific heat $c$.

• Heat Content: $Q(t) = \int_{x_1}^{x_2} c\rho u(x, t) dx$

• Rate of Change of Heat Content: $\frac{dQ}{dt} = \int_{x_1}^{x_2} c\rho \frac{\partial u}{\partial t} dx$

• Heat Flux: $\newline$

  • $\phi(x, t)$: heat flux, the rate of heat flow to the right per unit area.
  • $\phi(x, t) > 0$: heat flows to the right.
  • $\phi(x, t) < 0$: heat flows to the left.

• Conservation of Energy:

$\frac{dQ}{dt} = \phi(x_1, t) - \phi(x_2, t)$

$\int_{x_1}^{x_2} c\rho \frac{\partial u}{\partial t} dx = - \int_{x_1}^{x_2} \frac{\partial \phi}{\partial x} dx$

• Fourier's Law of Heat Conduction:

$\phi(x, t) = -K_0 \frac{\partial u}{\partial x}$

$K_0 > 0$: thermal conductivity.

• The Heat Equation: $\newline$ $c\rho \frac{\partial u}{\partial t} = \frac{\partial}{\partial x} (K_0 \frac{\partial u}{\partial x})$ $\newline$ If $K_0$ is constant: $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$ $\newline$ where $k = \frac{K_0}{c\rho}$ is the thermal diffusivity.

Initial Boundary Value Problems

• Initial Condition: $u(x, 0) = f(x)$

• Boundary Conditions:

$\newline$

• Prescribed Temperature:

$\newline$ . $u(0, t) = T_1$

$\newline$ . $u(L, t) = T_2$

$\newline$

• Insulated Boundary:

$\newline$ . $\frac{\partial u}{\partial x}(0, t) = 0$

$\newline$ . $\frac{\partial u}{\partial x}(L, t) = 0$ $\newline$

• Newton's Law of Cooling:

$\newline$ . $-\frac{\partial u}{\partial x}(0, t) = H[u(0, t) - T]$

$\newline$ . $\frac{\partial u}{\partial x}(L, t) = H[u(L, t) - T]$

$\newline$

• where $H > 0$ and $T$ is the ambient temperature.

Derivation of Steady State Temperature

• Steady-state temperature distribution $u(x)$ satisfies:

$0 = \frac{d}{dx}(K_0 \frac{du}{dx})$

• If $K_0$ is constant: $0 = \frac{d^2 u}{dx^2}$

• General solution: $u(x) = Ax + B$

• Apply boundary conditions to determine $A$ and $B$.

• Example: $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$ $\newline$

  • $u(0, t) = T_1$ $\newline$
  • $u(L, t) = T_2$ $\newline$
  • $u(x, 0) = f(x)$

• Steady-state solution: $u(x) = Ax + B$

• Applying boundary conditions: $\newline$

  • $u(0) = T_1 = A \cdot 0 + B \implies B = T_1$ $\newline$
  • $u(L) = T_2 = AL + T_1 \implies A = \frac{T_2 - T_1}{L}$

• Thus, $u(x) = \frac{T_2 - T_1}{L} x + T_1$