Download Lecture 5 Scaling and General Considerations

6/26/11 Lecture 5
Conditions at a discontinuity
Dimensionless parameters
1 Conservation conditions at an interface
The time rate of a change of a quantity ϕ(x, t) over a given volume V, moving
with velocity VI is1
�
�
�
d
∂ϕ
ϕdV =
dV + ϕ(VI · n) dS
dt V
V ∂t
S
Applied to mass conservation (ϕ = ρ) we
�
�
�
d
∂ρ
ρdV =
dV + ρ(VI · n) dS
dt V
V ∂t
S
Using the continuity equation
� �
V
�
∂ρ
+ ∇ · ρv dV = 0
∂t
d
dt
�
V
or
ρdV = −
�
S
�
V
∂ρ
dV +
∂t
�
S
ρ(v · n) dS = 0
ρ(v − VI )·n dS
1
This is the multidimensional analogue of Leibnitz’s theorem from calculus; it
is a kinematic relation, not a law of fluid mechanics.
2 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 1 6/26/11 When the control volume shrinks to the surface SI
� �
�
� �
�
d
+
−
lim
ρdV = −
ρ+(v −VI )·n+ + ρ−(v −VI )·n− dS
V→0 dt V
SI
�
��
�
=0
v − VI
no mass accumulation or
source/sink of mass at the interface.
n+ = n
−
+
ρ+(v · n − VI ) − ρ−(v · n − VI )
n−
[ ρ(v·n−VI ) ] = 0
“pillbox” control volume
VI is the velocity of the interface (control surface)
VI = VI ·n
[ pn + ρ(v·n−VI )v − Σ · n ] = 0
results from the momentum equation,. Similarly we can derive jump relations
for energy and species concentrations.
In fluid mechanics, discontinuities are allowed within the continuum framework,
provided the variables across the surface of discontinuity are such as to satisfy
the fundamental conservation laws, or the appropriate jump conditions.
3 If viscosity is negligible
[ ρ(v·n−VI ) ] = 0
[ pn + ρv(v·n−VI ) ] = 0
�
�
�
�
ρ h + 12 v 2 (v·n−VI ) + q·n = 0
[ ρYi ((v + Vi )·n−VI ) ] = 0
The momentum jump can be decomposed into normal and tangential components, to give
[ρ(v·n−VI )] = 0
[p + ρ(v·n−VI )(v·n)] = 0
[n × (v × n] = 0
Rankine-Hugoniot relations
4 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 2 6/26/11 Non-dimensional Equations
In the following the chemistry will be represented by a global one-step (irreversible) reaction
νF Fuel + νO Oxidizer → Products
describing the combustion of a single fuel.
The reaction will be assumed of order nF , nO , with respect to the fuel/oxidizer,
and an overall order n = nF + nO . The reaction rate will be assumed to obey
an Arrhenius law
ω=B
�
ρYF
WF
�nF�
ρYO
WO
�nO
e−E/RT
with an overall activation energy E and a pre-exponential factor B (treated as
constant).
5 For simplicity, we will treat µ, λ, ρDi constants (although most of the theoretical
development could accommodate a temperature-dependent λ) so that
∂ρ
+ ∇ · ρv = 0
∂t
�
�
Dv
ρ
= −∇p + µ ∇2 v + 13 ∇(∇ · v) + ρg
Dt
ρ
DYi
− ρDi ∇2 Yi = −νi Wi ω,
Dt
ρcp
i = F, O
DT
Dp
− λ∇2 T =
+ Φ + Qω
Dt
Dt
p = ρRT /W
ω=B
�
ρYF
WF
�nF�
ρYO
WO
�nO
e−E/RT
6 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 3 6/26/11 Characteristic values:
the fresh unburned state p0 , ρ0 , T0 (satisfying p0 = ρ0 RT0 /W ) for p, ρ, T .
a characteristic speed v0 to be specified
the diffusion length lD ≡ λ/ρcp v0 for distances
the diffusion time lD /v0 for t
This choice is clearly not unique and there may be other length, time, and
velocity scales that, for a given problem, could be more relevant.
We will use the same variables for the dimensionless quantities; i.e. after substituting v ∗ = v/v0 say, we remove the ∗ superscript.
7 ∂ρ
+ ∇ · ρv = 0
∂t
ρ
�
�
Dv
1
=−
∇p + Pr ∇2 v + 13 ∇(∇ · v) + Fr−1 ρeg
Dt
γM 2
DYF
2
− Le−1
F ∇ YF = −ω
Dt
DYO
2
ρ
− Le−1
O ∇ YO = −νω
Dt
�
�
DT
γ −1 Dp
2
2
ρ
−∇ T =
+ γP rM Φ + q ω
Dt
γ
Dt
ρ
p = ρT
ω = D ρn YFnF YOnO e−β0 /T
8 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 4 6/26/11 Dimensionless Parameters
v0
M= �
γp0 /ρ0
Pr =
µcp
λ
Mach number
Prandtl number
Fr =
Lei =
−1
Note that Pr = ρ0 v0 lD /µ = Re where Re is
the Reynolds number based on the diffusion length.
The Reynolds number based on a hydrodynamic
length L will be large because typically L � lD .
q=
Q/νF WF
c p T0
heat release
v02 /lD
g
λ/ρcp
Di
β0 = E/RT0
ν=
ν O WO
ν F WF
Froude number
Lewis number
of species i
Activation energy
mass weighted
stoichiometric coeff.
numerator is the heat release
per unit mass of fuel
D=
(lD /v0 )
[(ρ0 /WF )nF −1 (ρ0 /WO )nO B]−1
Damköhler number
=
flow time
reaction time
9 Low Mach Number Approximation
The propagation speed of ordinary deflagration waves is in the range 1-100
cm/s, namely much smaller than the speed of sound (in air a0 = 34, 000 cm/s).
M �1
momentum equation
⇒
∇p = 0
p = P (t) + γM 2 p̂(x, t) + · · ·
will all other variables expressed as v + γM 2 v̂ + · · ·
ρ
�
�
Dv
= −∇p̂ + Pr ∇2 v + 13 ∇(∇ · v) + Fr−1 ρeg
Dt
DT
γ −1 dP
ρ
− ∇2 T =
+qω
Dt
γ dt
ρT = P (t)
acoustic disturbances travel infinitely fast, and are filtered out.
10 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 5 6/26/11 Unless P (t) is specified, we are missing an equation, since p has been replaced
by two variables P and p̂. An equation in bounded problems can be obtained
as follows:
ρ
∂T
γ −1 dP
+ ρv · ∇T − ∇2 T =
+qω
∂t
γ dt
∂ρ
+ ∇ · ρv = 0
/T
∂t
∂(ρT )
γ −1 dP
+ ∇ · (ρvT ) − ∇2 T =
+qω
∂t
γ dt
1 dP
= −∇ · (P v − ∇T ) + qω
γ dt
1
γ
�
V
dP
dV = −
dt
�
S
(P v−∇T ) · n dS + q
�
ωdV
V
dP
γq
=
dt
V
on the surface S, v · n = 0 and for
adiabatic conditions ∂T /∂n = 0.
�
ω dV
V
11 The low Mach number equations are (with the ”hat” in p removed), therefore
∂ρ
+ ∇ · ρv = 0
∂t
�
�
Dv
ρ
= −∇p + Pr ∇2 v + 13 ∇(∇ · v) + Fr−1 ρeg
Dt
DYF
2
ρ
− Le−1
F ∇ YF = −ω
Dt
DYO
2
ρ
− Le−1
O ∇ YO = −νω
Dt
DT
γ − 1 dP
ρ
− ∇2 T =
+qω
Dt
γ dt
ρT = 1
and when the underlying pressure does not change in time, P = 1.
Unless otherwise indicated, we will be mostly concerned in the following with
this set of equations.
12 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 6 6/26/11 Coupling Functions
DYF
2
− Le−1
F ∇ YF = −ω
Dt
DYO
2
ρ
− Le−1
O ∇ YO = −νω
Dt
DT
ρ
− ∇2 T = q ω
Dt
ρ
For unity Lewis numbers the operator
on the left hand side of these three equations
is the same.
The combinations HF = T + qYF and HO = T + q YO /ν (and hence YF − YO /ν)
satisfy reaction-free equations
ρ
DHi
− ∇ 2 Hi = 0
Dt
leaving only one equation with the highly nonlinear reaction rate term. This is
a great simplification, but as we shall see, small variations of the Lewis numbers
from one produce instabilities and nontrivial consequences.
13 The constant-density approximation
ρT = 1
∂ρ
+ ∇ · ρv = 0
∂t
�
�
∂v
ρ(
+ (v · ∇)v) = −∇p + Pr ∇2 v + 13 ∇(∇ · v) + Fr−1 ρeg
∂t
solve to determine v
∂YF
2
+ v · ∇YF ) − Le−1
F ∇ YF = −ω
∂t
∂YO
2
ρ(
+ v · ∇YO ) − Le−1
O ∇ YO = −νω
∂t
∂T
ρ(
+ v · T ) − ∇2 T = q ω
∂t
ρ(
with the given v solve for T, YF , YO .
often referred to as the diffusive-thermal model
14 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 7