CFD Math simplified(how to do CFD)
Fluid Mechanics consists of
- Fluid Statics
- Fluid Dynamics
CFD is used to predict fluid flow using:
Conservation laws, Numerical methods, Algorithms
Steps involved in simulations:
- Physical model — describe physical model
- Mathematical model — describe equations that correspond with physical model
- Numerical methods — describe numerical methods that correspond with physical model and implement in code
- Geometry/Grid — describe grid that corresponds with physical model and implement in code
- Numerical solver — describe solver that corresponds with physical model and implement in code and compute solution
- Verification — establish solution validity
- Validation — compare with experimental data
Continuum Model
- Discontinuous molecular collisions are replaced with continuous statistically averaged values. Pressure = Average collision force per unit area.
- Continuum model enables conservation of mass, momentum and energy.
- Mean Free Path: the average distance that a moving particle travels between successive collisions or impacts.
- Knudsen number: the ratio of the molecular mean free path length to a representative physical length scale.
Continuum model is valid for Knudsen number << 1
For those who missed math classes
- Differential equations describe a point or particle and are needed for field information ie continuous problems.
- Integral equations describe control volume and fluxes across control surfaces and are needed for average quantities ie discontinuous problems.
- Degree of an equation: highest power
Order of an equation: highest derivative - ODE: relating a dependent variable (say x) to one independent variable (say u). derivative notation e.g. 𝑑𝑥/𝑑𝑡=𝑎 and 𝑑𝑢/𝑑𝑡=0
PDE: relating a dependent variable (say u) to more than one independent variable (say x and t). derivative notation e.g. ∂𝑢/∂𝑡+𝑎(∂𝑢/∂𝑥)=0 - A tensor is a multidimensional array
- Tensor product of two vectors:
Streamlines and Streaklines
Pressure
It is the normal force per unit surface. It is measured as absolute(wrt vacuum) and relative(wrt atmosphere).
- Static/thermodynamic: as a result of fluid’s temperature or internal energy.
- Hydrostatic: due to fluid column height. 𝑝_ℎ𝑦𝑑𝑟𝑜𝑠𝑡𝑎𝑡𝑖𝑐=𝜌𝑔ℎ
- Dynamic: due to fluid velocity. 𝑝_𝑑𝑦𝑛𝑎𝑚𝑖𝑐=1/2ρu2
- Stagnation: the pressure that would be reached if flow is brought to rest frictionlessly = static+ dynamic
- Total: sum of all components = static+ hydrostatic+ dynamic
Streamline is instantaneous tangent to the velocity field throughout the flow field. It changes with time in unsteady flow.
Pathline is trajectory of a fluid particle of fixed identity over a period of time. In unsteady flow pathlines cannot change their time history.
Kinematics
Kinematics: Quantities involving space and time only such as
- Position
- Velocity
- Acceleration
- Deformation
- Rotation
geometry of fluid motion
Dynamics:
- Refers to the stresses and forces that cause fluid motion
conservation laws
Lagrangian vs Eulerian approach
Lagrangian:
- Moving frame of reference
- Elements are matter
- Results in pathlines
- Advantage: More suitable than Euler for the dynamics of discrete particles in a fluid e.g. flow visualisation.
- Disadvantage: Computationally expensive to keep track of large numbers of particles in a flow field
Eulerian:
- Stationary frame of reference
- Elements are spatial
- Results in streamlines
- Advantage: More suitable than Lagrange for the dynamics of a fluid flow field, e.g. CFD
- Disadvantage: Time step limited by grid due to stability or accuracy
Integral and Derivative form
Conservation of mass
Cauchy momentum equation
Euler momentum equation
Navier Stokes momentum equation
The laws of physics are always expressed in the language of differential equations.
Solution to a differential equation is another function. The solution to algebraic function is a single value.
Finite element method predicts the behaviour of a component under certain circumstances. It uses differential equations to predict the rate of change of geometry temperature et cetera. we take our system and divided into finite elements. Now instead of working with one differential equation views element wise approximations.
Finite volume methods are conservative they respect the system is continuity.
