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1. INTRODUCTION TO CFD SPRING 2012
1.1 What is computational fluid dynamics? 1.2 Basic principles of CFD 1.3 Forms of the governing fluid-flow equations 1.4 The main discretisation methods Appendix
Examples
1.1 What is Computational Fluid Dynamics?
Computational fluid dynamics (CFD) is the use of computers and numerical methods to solve problems involving fluid flow.
CFD has been successfully applied in many areas of fluid mechanics. These include aerodynamics of cars and aircraft, hydrodynamics of ships, flow through pumps and turbines, combustion and heat transfer, chemical engineering. Applications in civil engineering include wind loading, vibration of structures, wind and wave energy, ventilation, fire and explosion hazards, dispersion of pollutants and effluent, wave loading on coastal and offshore structures, hydraulic structures such as weirs and spillways, sediment transport. More specialist CFD applications include ocean currents, weather forecasting, plasma physics, blood flow, heat dissipation from electronic circuitry.
This range of applications is very broad and involves many different fluid phenomena. In particular, the CFD techniques used for high-speed aerodynamics (where compressibility is significant but viscous effects are often unimportant) are very different from those used to solve low-speed, frictional flows typical of mechanical and civil engineering.
Although many elements of this course are generally applicable, the focus will be on simulating viscous , incompressible flow by the finite-volume method.
1.2 Basic Principles of CFD
The approximation of a continuously-varying quantity in terms of values at a finite number of points is called discretisation.
The fundamental elements of any CFD simulation are:
(1) The flow field is discretised ; i.e. field variables ( , u , v , w , p , โฆ) are approximated by their values at a finite number of nodes.
(2) The equations of motion are discretised (approximated in terms of values at nodes): control-volume or differential equations algebraic equations ( continuous ) ( discrete )
(3) The resulting system of algebraic equations is solved to give values at the nodes.
The main stages in a CFD simulation are: Pre-processing :
- formulation of the problem (governing equations and boundary conditions);
- construction of a computational mesh (set of nodes and control volumes). Solving :
- discretisation of the governing equations;
- solution of the resulting algebraic equations. Post-processing :
- visualisation (graphs and plots of the solution);
- analysis of results (calculation of derived quantities: forces, flow rates, ... )
1.3 Forms of the Governing Fluid-Flow Equations
The equations governing fluid motion are based on fundamental physical principles:
- mass : change of mass = 0
- momentum : change of momentum = force ร time
- energy : change of energy = work + heat
Additional equations may apply for non-homogeneous fluids (e.g. containing dissolved chemicals or imbedded particles).
When applied to a fluid continuum all these conservation equations may be expressed mathematically in equivalent:
- integral (i.e. control-volume );
- differential forms.
continuous curve
discrete approximation
1.4 The Main Discretisation Methods
(i) Finite-Difference Method
Discretise the governing differential equations; e.g.
y
v v x
u u y
v x
u (^) i j i j ij ij โ
0 1 ,^1 , ,^1 ,^1
(ii) Finite-Volume Method
Discretise the governing integral or control-volume equations; e.g. netmassoutflow = ( uA ) e โ( uA ) w +( vA ) n โ( vA ) s = 0
(iii) Finite-Element Method
Express the solution as a weighted sum of shape functions S ^ ( x ); e.g., for velocity:
u ( x ) = โ u S ( x )
Substitute into some form of the governing equations and solve for the coefficients (aka
degrees of freedom or weights ) u ^.
Finite-difference and finite-element methods are covered in more detail in the Computational Mechanics course. This course will focus on the finite-volume method.
The finite-element method is popular in solid mechanics (geotechnics, structures) because:
- it has considerable geometric flexibility;
- general-purpose codes can be used for a wide variety of physical problems.
The finite-volume method is popular in fluid mechanics because:
- it rigorously enforces conservation ;
- it is flexible in terms of both geometry and the variety of fluid phenomena ;
- it is directly relatable to physical quantities (mass flux, etc.).
In the finite-volume method ...
(1) A flow geometry is defined.
(2) The flow domain is decomposed into a set of control volumes or cells called a computational mesh or grid.
(3) The control-volume equations are discretised โ i.e. approximated in terms of values at nodes โ to form a set of algebraic equations.
(4) The discretised equations are solved numerically.
i,j
i,j+
i,j-
i-1,j i+1,j
v
ue
vn uw
s
APPENDIX
A1. Notation
Position/time: x โก ( x , y , z ) or ( x 1 , x 2 , x 3 ) position; (in this course z is usually vertical) t time Field variables: u โก ( u , v , w ) or ( u 1 , u 2 , u 3 ) velocity p pressure ( p โ p atm is the gauge pressure ; p * = p + gz is the piezometric pressure .) T temperature ฯ concentration (amount per unit mass or per unit volume) Fluid properties: density dynamic viscosity ( โก / is the kinematic viscosity) diffusivity
A2. Mechanical Principles
Statics
At rest, pressure forces balance weight. This can be written mathematically as
p = โ g z or g z
p d
d = โ (3)
The same equation also holds in a moving fluid if there is no vertical acceleration, or, as an approximation, if vertical acceleration is much smaller than g.
If density is constant, (3) can be written p * โก p + gz = constant (4)
p * is called the piezometric pressure , combining the effects of pressure and weight. For a constant-density flow without a free surface, gravitational forces can be eliminated entirely from the equations by working with the piezometric pressure.
Pressure, density and temperature are connected by an equation of state ; e.g. ideal gas law : p = RT , R = R 0 / m (5)
where R 0 is the universal gas constant, m is the molar mass and T is the absolute temperature.
Dynamics
The equations governing fluid motion are based on the following fundamental principles:
- mass : change of mass = 0
- momentum : change of momentum = force ร time
- energy : change of energy = work + heat and for non-homogeneous fluids:
- conservation of individual constituents.
