Getting Started

1.1. Getting Started#

This is a very quick overview of some fundamental knowledge that is helpful for understanding the analytic and numeric work needed for FVM. Some of it is calculus and continuum mechanics that should be mostly review (and can be used as a reference for notation), whereas the later sections give some context and insight into the derivation of commonly used conservation laws (PDE’s) that can be solved using FVM. A few selected LaTeX equations are provided as well to help you use the same notation in your Markdown-formatted reports (optional).

Notation#

A generic scalar quantity \(\phi\) is typically defined in 3-dimensional (3D) space and time, \(\phi(x,y,z,t)\).

Bold straight lower-case letters are used for vector quantities. For example, the Cartesian coordinate system is:

\[ \mathbf{x} = x \,\hat{i} + y \,\hat{j} + z \,\hat{k} \]

The scalar quantity defined above could thus be specified \(\phi(\mathbf{x},t)\). Velocity is denoted \(\mathbf{u}\), with components \(u\), \(v\) and \(w\):

\[ \mathbf{u} = u \,\hat{i} + v \,\hat{j} + w \,\hat{k} \]

In the finite volume discretization scheme, volume indices are denoted with a subscript that refers to increments of each axis in \(\mathbf{x}\), starting at index 1 (not 0, as in Python!). For example, subscript \(i\), \(j\) are used in 2D, such that the volume centered as \(x\)-coordinate 4 and \(y\)-coordinate 6 would be described with velocity \(\mathbf{u}_{4,6}^n\). The time increment is identified with a superscript, \(n\); the initial condition starts at index 0. In other words, the initial condition is specified at time \(t_0\).

Locations or quantities in a 1-dimensional (1D) discretized continuum in are expressed as a column vector, for example, the velocity in the \(x\)- or \(\hat{i}\)-direction at points \(i=1:n\) is:

\[\begin{split} u_i \; \forall \; i = \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{bmatrix} = \begin{bmatrix} u_1 & u_2 & \dots & u_n\end{bmatrix}^T \end{split}\]

Selected LaTeX equations:

$$
\mathbf{u} = u \,\hat{i} + v \,\hat{j} + w \,\hat{k}
$$

Derivatives#

The total or direct derivative uses \(d\). For example, if \(z=g(y)\) and \(y=f(x)\), the chain rule is:

\[ \frac{dz}{dx} = \frac{d}{dx}g(f(x)) = \frac{dg}{df}\frac{df}{dx} \]

The partial derivative uses \(\partial\). For example, the time derivative of \(f(\mathbf{x},t)\) can be represented as follows:

\[ \frac{\partial f}{\partial t} = \frac{\partial f(\mathbf{x},t)}{\partial t} \]

The material derivative is defined using an uppercase letter \(D\) to denote the direct derivative in the Lagrangian frame:

\[ \frac{D\mathbf{u}}{Dt} = \frac{\partial\mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\,\mathbf{u} \]

Selected LaTeX equations:

$$
\frac{D\mathbf{u}}{Dt}
= \frac{\partial\mathbf{u}}{\partial t}
+ (\mathbf{u}\cdot\nabla)\,\mathbf{u}
$$

Nabla Operations#

The nabla operator is:

\[ \nabla\ = \frac{\partial }{\partial x} \,\hat{i} + \frac{\partial }{\partial y} \,\hat{j} + \frac{\partial }{\partial z} \,\hat{k} \]

When applied to a scalar, \(\nabla\) specifies the gradient, which produces a vector quantity:

\[ \nabla\phi = \frac{\partial \phi}{\partial x} \,\hat{i} + \frac{\partial \phi}{\partial y} \,\hat{j} + \frac{\partial \phi}{\partial z} \,\hat{k} \]

When applied to a vector, \(\nabla\) specifies the divergence, which produces a scalar quantity:

\[ \nabla\cdot\mathbf{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \]

When a vector is applied to \(\nabla\) the div operator is specified, which produces a scalar quantity that is typically applied to another scalar, vector or tensor as part of a larger expression:

\[ \mathbf{u}\cdot\nabla = u\frac{\partial}{\partial x} + v\frac{\partial}{\partial y} + w\frac{\partial}{\partial z} \]

The Laplacian is defined as \(\nabla^2\), and is the divergence of the gradient \(\nabla \cdot \nabla\). When applied to a scalar it produces a scalar quantity:

\[ \nabla^2 \phi = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \]

Selected LaTeX equations:

$$
\nabla\
= \frac{\partial }{\partial x} \,\hat{i}
+ \frac{\partial }{\partial y} \,\hat{j}
+ \frac{\partial }{\partial z} \,\hat{k}
$$

Material Derivative#

The material derivative (also called Lagrangian derivative) describes the time rate of change of a scalar or vector quantity \(\phi(\mathrm{x},t)\) that is subject to a velocity field \(\mathrm{u}(\mathrm{x},t)\):

\[ \frac{\mathrm{D} \phi}{\mathrm{D}t} = \frac{\partial \phi}{\partial t} + \mathbf{u} \cdot \nabla \phi \]

where \(\mathrm{D}\) represents the total derivative in the Lagrangian reference frame.

Conservation Laws#

Most physics-based modelling relies on the use of conservation laws. Most common are mass, momentum, energy, transport, here mas conservation and the equation of transport are described.

Mass#

Mass conservation is defined using the continuity equation, which describes the flux of a quantity per unit volume:

\[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho\mathbf{u}) = 0 \]

where \(\rho\) and \(\mathbf{u}\) are the density and velocity of a material; for the FVM we will typically consider fluids, but the law is also valid for (deformable) solids.

Equation of Transport#

Many phenomena can be described by the Equation of Transport, here formulated using the material derivative. Consider the quantity to be a scalar or vector field and a function of space and time, \(\phi=\phi(\mathbf{x}, t)\). This quantity of interest can be a scalar like a Temperature or a polutant, a vector flow velocity, etc. So for a fluid, its particles are the ones transporting the quantity \(\phi\), which is described by a velocity vector field, \(\mathbf{u}\) (indeed, the flow velocity transports itself: momentum). The general form reads:

\[ \frac{\partial(\rho\phi)}{\partial t} + \nabla \cdot (\rho\phi\mathbf{u}) = \nabla \cdot (D \nabla \phi) +F_{\phi} \]

For incompressible flow, constant diffusion coefficient (\(D\)) and accounting for mass conservation, the equation can be expressed as:

\[ \frac{\partial \phi}{\partial t} + \mathbf{u} \cdot \nabla \phi = D^* \nabla^2 \phi +F^*_{\phi} \]

where \(D^*\) and \(F^*\) are \(D\) and \(F\) divided by \(\rho\). From now on, the asterisk will be dropped.

From left to right, these three terms are explained as follows, relative to the quantity \(\phi\):

  1. Transient effect: rate of change of quantity \(\phi\)

  2. Convection: transport due to velocity field \(\mathbf{u}\)

  3. Diffusion: transport due to gradient of quantity \(\phi\)

  4. Fount/Source \(F_\phi\): it refers to a quantity that can be produced or removed locally. For example, a point pollution source.

Sometimes the term convection is used interchangeably with advection; technically the difference is that advection specifies a quantity that is carried by the fluid, rather than the fluid particles themselves (convection).

Selected LaTeX equations:

$$
\frac{\partial \phi}{\partial t}
+  \mathbf{u} \cdot \nabla \phi
= D \nabla^2 \phi + S_{\phi}
$$

PDE’s nature#

In the previous numerical modelling weeks, we saw the classification of differential equations in ODEs, PDEs and their order. Now we will consider the physical nature of PDEs as it will give insight into the correct modeling approach. We dealt with the temperature distribution along a rod when the problem was steady (in equilibrium) and when it was unsteady.

Elliptic equations are generally steady state problems just like the Temperature distribution in equilibrium along a rod, or stress distribution in a solid object due to a load. A relevant feature is that a disturbance in the interior points, e.g., a single one, changes the solution in the entire domain. So, the information propagates instantly in every direction. A classic elliptic equation is Laplace

\[ \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} = 0 \]

In Parabolic equations, the propagation speed of information is also infinite. This kind of problems are dominated by diffusion, so the solutions tend to be smooth. A classic parabolic equation is the diffusion equation (here shown in 1D):

\[ \frac{\partial\phi}{\partial t} = D\frac{\partial^2\phi}{\partial x^2} \]

In Hyperbolic equations, the information travels at a finite speed, i.e. the influence of a local perturbation is also local! Due to the lack of a diffusion mechanism, discontinuities (or sharp local gradients in \(\phi\)) persist!. A classic hyperbolic equation is the advection equation (here shown in 1D):

\[ \frac{\partial\phi}{\partial t} + c \frac{\partial\phi}{\partial x} = 0 \]

where the quantity \(\phi\) is transported by the fluid velocity \(u\).

Note that these equations can be derived from the Equation of transport, in the absence of source terms and a solid (no advection) the diffusion equation is obtained. In the absence of source terms and diffusion, the advection equation is obtained.

Exercises

Attribution

This chapter is written by Robert Lanzafame and Jaime Arriaga Garcia. Find out more here.

Contents