Symbols¶
List of symbols that will be referenced in Theory, Tutorials and Case Studies.
General Variables¶
\(q, s\) \(~~~~~~~~~~~\) Scalars
\(\mathbf{a}, \mathbf{b}, \mathbf{c}\) \(~~~~~~\) Vectors
\(\mathbf{S}, \mathbf{T}, \mathbf{Y}\) \(~~~~\) Tensors
\(x, y, z\) \(~~~~~~~\) Cartesian Coordinate System
\(\mathbf{I}\) \(~~~~~~~~~~~~~~~\) Identity Tensor
\(\mathbf{\Psi}\) \(~~~~~~~~~~~~~\) General Parameter: scalar, vector or tensor
Geometry and Mesh¶
\(\mathbf{C}\) \(~~~~~~\) Cell Centre
\(\mathbf{C_f}\) \(~~~~~\) Face Centre
\(\mathbf{n}\) \(~~~~~~~\) Normal Vector of Unit Length
\(\mathbf{p}\) \(~~~~~~~\) Point Location
\(S\) \(~~~~~~~\) Surface
\(V\) \(~~~~~~~\) Volume
Numerical Methods¶
\([\mathbf{A}]\) \(~~~~~~\) Matrix of a Matrix Equation \([\mathbf{A}][\mathbf{\Psi}]=[\mathbf{B}]\)
\(\mathbf{A}\) \(~~~~~~~~\) Matrix of a Matrix Equation \(\mathbf{A} \cdot \mathbf{\Psi}=\mathbf{B}\)
\(a_{i,j}\) \(~~~~~~\) Matrix Coefficients for row \(i\) and column \(j\)
\([\mathbf{b}]\) \(~~~~~~~\) Source Vector of a Matrix Equation
\(\mathbf{b}\) \(~~~~~~~~~\) Source Vector of a Matrix Equation, above
\(c\) \(~~~~~~~~~~\) Scalar Coefficients
MORE TO ADD
Subscripts¶
\(b\) \(~~~~~~\) Boundary Value/Gradient
\(D\) \(~~~~~\) Downwind Interpolation
\(ex\) \(~~~~\) Exact Solution
\(f\) \(~~~~~~\) Values at Faces
\(L\) \(~~~~~~\) Linear Interpolation, Lower Bound Value
\(N\) \(~~~~~\) Neighbour Cell
\(P\) \(~~~~~~\) Owner/Current Cell
\(U\) \(~~~~~~\) Downwind Interpolation
Superscripts¶
\(c\) \(~~~~~~\) Current Stored Value
\(n\) \(~~~~~~\) New Value
\(o\) \(~~~~~~\) Old Time Level
\(oo\) \(~~~~\) Old-Old Time Level
Dimensionless Numbers¶
\(Co\) \(~~~~\) Courant Number = \((\Delta t/{V})\Sigma_f \cdot \phi^+\)
\(Pe\) \(~~~~\) Peclet Number = \(\phi_f~/ (|\mathbf{S_f}|~\alpha_f~C_{\Delta})\)
Physical Quantities (SI units)¶
\(\mathbf{b}\) \(~~~~\) Body Force per Unit Mass \((N \cdot kg^{-1})\)
MORE TO ADD
Dimensionless Numbers¶
\(Eu\) \(~~~~\) Euler Number = \(p_0~/U^2\)
\(Fr\) \(~~~~\) Froude Number = \(U~/ \sqrt{gL}\)
\(Kn\) \(~~~\) Knudsen Number = \(\lambda~/ L\)
\(Ma\) \(~~~\) Mach Number = \(|\mathbf{u}|~/c\)
\(Pe\) \(~~~~\) Peclet Number = \(U L~/ \alpha\)
\(Pr\) \(~~~~\) Prandtl Number = \(\nu~/\alpha\)
\(Re\) \(~~~~\) Reynolds Number = \(U L~/ \nu\)
\(Re_\tau\) \(~~~\) Turbulent Reynolds Number = \(k^2~/ \epsilon\nu\)
\(Sc\) \(~~~~\) Schidt Number = \(\nu~/D\)
\(Sr\) \(~~~~\) Strouhal Number = \(L~/(\mathcal{T} U)\)
Relational Symbols¶
\(a=b\) \(~~~~\) \(a\) and \(b\) are equal
\(a \approx b\) \(~~~~\) \(a\) and \(b\) are approximately equal
\(a \simeq b\) \(~~~~\) \(a\) and \(b\) are almost exactly equal
\(a \equiv b\) \(~~~~\) \(a\) and \(b\) are equivalent
\(a:=b\) \(~~~\) \(a\) is assigned the value of \(b\)
\(a \propto b\) \(~~~~\) \(a\) is proportional to \(b\)
\(a \sim b\) \(~~~~\) \(a\) is of the order of magnitude of \(b\)
Credits¶
Weller, H. Greenshields C. (2022) Notes on Computational Fluid Dynamics: General Principles. Available at: CFD Direct, Notes on CFD (Accessed: 09 January 2025)