Computational Fluid Dynamics: The Finite-Volume Method...Computational Fluid Dynamics? The use of...
Transcript of Computational Fluid Dynamics: The Finite-Volume Method...Computational Fluid Dynamics? The use of...
Computational Fluid Dynamics:The Finite-Volume Method
David Apsley
1. Introduction
What is β¦Computational Fluid Dynamics?
The use of computers and
numerical methods to solve
problems involving fluid flow
Aerodynamics
Wind Loading
Turbine Technology
Vortex Shedding
Dispersion of Pollution
Ventilation
Particle-Laden Plumes (Sea Outfalls)
(ex302)ra=1040 kg/m3
(ex322)ra=1020 kg/m3
(ex336)ra=1020 kg/m3
Ua=0.026 m/s
free surface
D U0
0z
h
r0,
Uar a,
Ws
solid bed
Sediment Scour
River bend Bridge pier
Discretisation
Field variables:
Equations:
continuouscurve
discreteapproximation
f
x x
f
2
1
dπ
dπ₯β
Ξπ
Ξπ₯=
π2 β π1π₯2 β π₯1
Basic Principles of CFD
1. Discretise space:
replace field variables (r, π’, π£, π€, π, β¦) by values at a finite number of nodes
2. Discretise equations:
continuum equations β algebraic equations
3. Solve:
large system of simultaneous equations
Stages of a CFD Analysis
β’ Pre-processing:
β formulate problem (geometry, equations, boundary conditions)
β construct computational mesh
β’ Solving:
β discretise
β solve
β’ Post-processing:
β analyse
β visualise (graphs and plots)
Fluid-Flow Equations
β’ Mass: change of mass = 0
β’ Momentum: change of momentum = force Γ time
β’ Energy: change of energy = work + heat
β’ (Other constituents)
In fluid mechanics, these are normally expressed in rate form
Form of Equations
β’ Integral (control-volume)
β’ Differential
Integral (Control-Volume) Approach
β Finite-volume method for CFD
Consider the budget of any transported physical quantity in any control volume
V
CHANGE = IN β OUT + CREATED
CHANGE
time+
OUT β IN
time=
CREATED
time
TIME DERIVATIVEof amount in π
+NET FLUX
through boundary of π=
SOURCEinside π
TIME DERIVATIVEof amount in π
+ADVECTION + DIFFUSIONthrough boundary of π
=SOURCEinside π
Differential Equations For Fluid Flow
β’ Derived by considering the rate of change at a point; i.e. using infinitesimal control volumes
β’ Discretisation gives a finite-difference method for CFD
β’ Several types:
β fixed-point (βEulerianβ): conservative
β moving with the flow (βLagrangianβ): non-conservative
β derived variables; e.g. potential flow
Main Methods for CFD
β’ Finite-difference:
β discretise differential equations
β’ Finite-volume:
β discretise control-volume equations
β’ Finite-element:
β represent solution as a weighted sum of basis functions
i,j
i,j+1
i,j-1
i-1,j i+1,j
ew
v
nv
s
uu
0 =ππ’
ππ₯+ππ£
ππ¦
π’(x) = )π’Ξ±πΞ±(x
0 = net mass outflow
βπ’π+1,π β π’πβ1,π
2Ξπ₯+π£π,π+1 β π£π,πβ1
2Ξπ¦
= Οπ’π΄ π β Οπ’π΄ π€ + Οπ£π΄ π β Οπ£π΄ π
Advantages of the Finite-Volume Method in CFD
β’ Rigorously enforces conservation
β’ Flexible in terms of:
β geometry
β fluid phenomena
β’ Directly relatable to physical quantities
b
Examples
Example Q1
Water (density 1000 kg mβ3) flows at 2 m sβ1 through a circular pipe of diameter 10 cm. What is the mass flux πΆacross the surfaces π1 and π2?
2 m/s
10 c
m
S S12
45o
Water (density 1000 kg mβ3) flows at 2 m sβ1 through a circular pipe of diameter 10 cm. What is the mass flux πΆ across the surfaces π1 and π2?
2 m/s
10 c
m
S S12
45o
π1: mass flux (πΆ) = Οπ’π΄
= 1000 Γ 2 ΓΟ Γ 0. 12
4
= 15.7 kg sβ1
π2: the same!
In general: πΆ = Οu β’ A
πΆ = Ο(π’ cos ΞΈ)π΄
)πΆ = Οπ’(π΄ cos ΞΈ
Example Q2
A water jet strikes normal to a fixed plate as shown. Compute the force πΉ required to hold the plate fixed.
Fu = 8 m/s
D = 10 cm
A water jet strikes normal to a fixed plate as shown. Compute the force πΉ required to hold the plate fixed.
Fu = 8 m/s
D = 10 cm
force on fluid = momentum flux out β momentum flux in
momentum flux = mass flux Γ velocity
βπΉ = 0 β (Οππ΄)π
πΉ = Οπ2π΄
= 1000 Γ 82 ΓΟ Γ 0. 12
4
= 503 N
Example Q3
An explosion releases 2 kg of a toxic gas into a room of dimensions 30 m 8 m 5 m. Assuming the room air to be well-mixed and to be vented at a speed of 0.5 m sβ1 through an aperture of area 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.
(Take the density of air as 1.2 kg mβ3.)
An explosion releases 2 kg of a toxic gas into a room of dimensions 30 m 8 m 5 m. Assuming the room air to be well-mixed and to be vented at a speed of 0.5 m sβ1 through an aperture of area 6 m2, calculate:(a) the initial concentration of gas in ppm by mass;
π = 30 Γ 8 Γ 5 = 1200 m3
mass of fluid Γ concentration = mass of toxin
(Οπ)Ο0 = 2 kg
Ο0 =2
1.2 Γ 1200
Volume V
Concentration
U
area A
= 1.389 Γ 10β3
1389 ppm
An explosion releases 2 kg of a toxic gas into a room of dimensions 30 m 8 m 5 m. Assuming the room air to be well-mixed and to be vented at a speed of 0.5 m sβ1 through an aperture of area 6 m2, calculate:(b) the time taken to reach a safe concentration of 1 ppm.
π = 1200 m3
Volume V
Concentration
U
area Aπ΄ = 6 m2
π = 0.5 m sβ1
Ο0 = 1389 ppm
Change in amount of toxin = amount in β amount out
Rate of change of amount of toxin = rate of entering β rate of leaving
d
dπ‘(ΟπΟ) = 0 β (Οπ’π΄)Ο
dΟ
dπ‘= β
π’π΄
πΟ, Ο = Ο0 at π‘ = 0
dΟ
dπ‘= βΞ»Ο Ξ» =
π’π΄
π= 0.0025 sβ1
Ο = Ο0eβΞ»π‘
eΞ»π‘ =Ο0
Ο
π‘ =1
Ξ»lnΟ0
Ο=
1
0.0025ln 1389 = 2895 s
β 48 min