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1. INTRODUCTION TO CFD 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

SPRING 2011

1.1 What is Computational Fluid Dynamics? Computational fluid dynamics (CFD) is the use of computers and numerical techniques to solve problems involving fluid flow. CFD has been successfully applied in many areas of fluid mechanics, including: aerodynamics of cars and aircraft; hydrodynamics of ships; pumps and turbines; combustion and heat transfer; chemical engineering. Applications in civil engineering include: wind loading and dynamic response of structures; wind, wave and tidal 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, hydrology. More specialist applications include ocean currents, weather forecasting, plasma physics, blood flow, heat dissipation from electronic circuitry. This range of applications is broad and encompasses many different fluid phenomena and CFD techniques. In particular, CFD for high-speed aerodynamics (where compressibility is significant but viscous effects are often unimportant) is very different from that 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.

CFD

1–1

David Apsley

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.

continuous curve

discrete approximation

The fundamental simulation are: (1)

elements

of

any

CFD

The flow field is discretised; i.e. field variables ( , u, v, w, p, …) are approximated by their values at a finite number of nodes. The equations of motion are discretised (approximated in terms of values at nodes): control-volume or differential equations algebraic equations (continuous) (discrete) The resulting system of algebraic equations is solved to give values at the nodes.

(2)

(3)

The main stages in a CFD simulation are: Pre-processing: – problem formulation (governing equations; boundary conditions); – construction of a computational mesh. Solving: – discretisation of the governing equations; – numerical solution of the governing equations. Post-processing: – visualisation; – analysis of results.

1.3 Forms of the Governing Fluid-Flow Equations The equations governing fluid motion are based on the fundamental physical principles: mass: change of mass = 0 momentum: change of momentum = force × time energy: change of energy = work + heat Additional conservation equations for individual constituents may apply for nonhomogeneous fluids (e.g. containing dissolved chemicals or imbedded particles). When applied to the fluid continuum these may be expressed mathematically as either: integral (i.e. control-volume) equations; differential equations.

CFD

1–2

David Apsley

1.3 Forms of the Governing Fluid-Flow Equations 1.3.1 Integral (Control-Volume) Approach This considers changes to the total amount of some physical quantity (mass, momentum, energy, …) within a specified region of space (control volume).

V

For an arbitrary control volume the balance of a physical quantity over an interval of time is change = amount in amount out + amount created In fluid mechanics this is usually expressed in rate form by dividing by the time interval (and transferring the net amount passing through the boundary to the LHS of the equation): RATE OF CHANGE + NET FLUX = SOURCE (1) out of boundary inside V inside V The rate of transport across a surface or flux is composed of: advection – movement with the fluid flow; diffusion – net transport by random (molecular or turbulent) motion.

RATE OF CHANGE + ADVECTION + DIFFUSION = SOURCE through boundary of V inside V inside V

(2)

The important point is that this is a single, generic scalar-transport equation, irrespective of whether the physical quantity concerned is mass, momentum, chemical content etc. Thus, instead of dealing with lots of different equations we can consider the numerical solution of a general scalar-transport equation (Section 4). The finite-volume method, which is the subject of this course, is based on approximating these control-volume equations.

1.3.2 Differential Equations In regions without shocks, interfaces or other discontinuities, the fluid-flow equations can also be written in equivalent differential forms. These describe what is going on at a point rather than over a whole control volume. Mathematically, they can be derived by making the control volumes infinitesimally small. This will be demonstrated in Section 2, where it will also be shown that there are several different ways of writing these differential equations. The finite-difference method is based on the direct approximation of a differential form of the governing equations.

CFD

1–3

David Apsley

1.4 The Main Discretisation Methods

i,j+1

(i) Finite-Difference Method

i-1,j i,j i+1,j

Discretise the governing differential equations directly; e.g. u i +1, j u i 1, j vi , j +1 vi , j 1 u v ≈ + 0 = + x y 2x 2y

i,j-1

(ii) Finite-Volume Method Discretise the governing control-volume equations directly; e.g. net mass outflow = ( uA) e ( uA) w + ( vA) n ( vA) s

uw

vn ue vs

= 0

(iii) Finite-Element Method Express the solution as a weighted sum of shape functions S (x), substitute into some form of the governing equations and solve for the coefficients (aka degrees of freedom or weights). e.g., for velocity, u (x) = ∑ u S (x)

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 – i.e. approximated in terms of values at nodes – to form a set of algebraic equations. (4) The discretised equations are solved numerically.

CFD

1–4

David Apsley

APPENDIX A1. Notation Position/time: x ≡ (x, y, z) or (x1, x2, x3) position; (in this course z is usually vertical) t time Field variables: u ≡ (u, v, w) or (u1, u2, u3) velocity p pressure (p – patm is the gauge pressure; p* = p + gz is the piezometric pressure.) T temperature φ concentration (amount per unit mass or per unit volume) Fluid properties: density dynamic viscosity ( ≡ / is the kinematic viscosity) diffusivity

A2. Mechanical Principles Statics At rest, pressure forces balance weight. This can be written mathematically as dp = g (3) p= g z or dz 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* ≡ 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 = R0 /m (5) where R0 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 × time energy: change of energy = work + heat and for non-homogeneous fluids: conservation of individual constituents. CFD 1–5 David Apsley

Examples The following simple examples develop the notation and control-volume framework to be used in the rest of the course.

Q1. Water (density 1000 kg m–3) flows at 2 m s–1 through a circular pipe of diameter 10 cm. What is the mass flux C across the surfaces S1 and S2?

10 cm

2 m/s

45

o

S1

S2

D=10 cm

u=8 m/s

F

Q2. A water jet strikes normal to a fixed plate (see left). Neglect gravity and friction, and compute the force F required to hold the plate fixed.

Q3. A fire in the Pariser hydraulics laboratory released 2 kg of a toxic gas into a room of dimensions 30 m × 8 m × 5 m. Assuming the laboratory air to be well-mixed and to be vented at a speed of 0.5 m s–1 through an aperture of 6 m2, calculate: (a) the initial concentration of gas in ppm by mass; (b) the time taken to reach a safe concentration of 1 ppm. (For air, density = 1.20 kg m–3.)

Q4. A burst pipe at a factory causes a chemical to seep into a river at a rate of 2.5 kg hr–1. The river is 5 m wide, 2 m deep and flows at 0.3 m s–1. What is the average concentration of the chemical (in kg m–3) downstream of the spill?

CFD

1–6

David Apsley

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