Download Nondimensionalization of equations

Appendix L
Nondimensionalization of
equations
L.1
Flow around a ball
Let us try to integrate the dimensional analysis of a certain problem with the
information arising from the equations of motion and from other knowledges of
physical nature.
Let us consider the flow around a spherical ball moving at constant velocity
in a fluid. First of all, we know that there are two different dynamical regimes,
the laminar regime and the turbulent regime.
When the velocity of the ball is sufficiently small, so that the Reynolds number is very low, we obtain a laminar flow called creeping flow. As a certain
magnitude of the velocity in the far field is overcome, the nonlinear terms will
become dominant. For sufficiently large Reynolds numbers the flow will be fully
turbulent.
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L.2
Franco Mattioli (University of Bologna)
The equations for a creeping flow
In a creeping flow the motion, as seen from a reference system in which the
ball is at rest, is governed by the continuity equation
∇·u = 0
and by the momentum equation
1
0 = − ∇p + ν∇2 u,
ρ
where p is the pressure, u the velocity, ρ the density of the fluid and ν its
kinematic viscosity.
The space quantities can be nondimensionalized on the basis of the radius R
of the sphere and of the velocity U of the flow far from it, by setting
x = Rx0 ,
y = Ry 0 ,
z = Rz 0
u = U u0 .
and
(L.1)
In essence, we have assumed the radius of the sphere as the unit of measurement
of the lengths and the velocity far from the body as the unit of measurement of
the velocities. The momentum equation becomes
νU
R
2
∇02 u0 =
1 0
∇ p,
Rρ
where x0 , y 0 , z 0 and u0 are dimensionless quantities and ∇0 and ∇02 are respectively
the gradient and the laplacian in dimensionless coordinates
0
∇ =
∂ ∂ ∂
, ,
x0 y 0 z 0
!
and
2
∇0 =
∂2
∂x0
2
+
∂2
∂y 0
2
+
∂2
∂z 0
2.
(L.2)
By substituting to the pressure p the dimensionless variable p0 defined by the
relation
ρνU 0
p=
p,
(L.3)
R
then the equations of motion take the dimensionless form
∇0 · u0 = 0
and
2
∇0 u0 = ∇0 p0 .
Principles of Fluid Dynamics (www.fluiddynamics.it)
xci
Within the framework of the language adopted in the previous chapters, we might
have written
∇0 · η 1 = 0
and
2
∇0 η 1 = ∇0 η2 .
The dimensionless formulation maintain the same structure of the various terms of
the equations, but eliminates all the scalar parameters that appear in it. It is a formulation more elegant than the original one, but it contains the same level of mathematical
difficulties.
L.3
Drag force in a creeping flow
To move the sphere in the fluid, it is clearly necessary to exert a certain force.
Such a force is the so-called drag force D. This force is given by the integral of
the total stress over the whole surface 4πR2 of the sphere. The stress has the
same dimension of the pressure. It follows that the dimension of the force is given
by
#
"
ρνU
2
[R2 ] = [µ U R],
[D] = [p][R ] =
R
where µ = ρν. Once introduced the dimensionless variable
η1 =
D
,
µ UR
the solution of the problem will take the form
η1 = c.
In this case the equations of the motion are solvable by means of the traditional
(even if not too simple) methods of mathematics, which provide c = 6π, so that
η1 = 6π.
It follows the Stokes law for a creeping flow
D = 6πµRU.
(L.4)
Thus, not only the equation of motions can be made dimensionless, but the
dimensional structure of the solution can also be derived. In this (dimensionally)
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Franco Mattioli (University of Bologna)
simple case all we have to do is to define a trivial multiplicative constant, which
might be derived by a single experiment.
In more complicated problems we might have more parameters to define. If in place of
a spherical ball we had an ellipsoidal body of rotation, then instead of a single parameter,
the radius of the sphere, we would have two parameters, that is, the two semi-axes of
the ellipsoid.
L.4
Terminal velocity of a ball sinking in a fluid
Let us now consider a slightly more complicated problem. We will try to
evaluate the asymptotic velocity of a sphere immersed in a fluid and subject to
the action of the gravity field, under the hypothesis of creeping flow. The weight
P of the sphere is equal to
4
(L.5)
P = πR3 ρs g.
3
where ρs is the density of the material of which the sphere is made. Once introduced the dimensionless variable
η2 =
P
,
R ρs g
we have
η2 =
3
4
π.
3
When the asymptotic regime is reached, the weight P must be balanced by
the drag force D, so that, by equating (L.4) and (L.5), we have
η3 =
η2
µU
2
= 2
= ,
η1
9
R ρs g
(L.6)
from which the solution of our problem can be attained
U=
2 R2 g ρs
.
9 ν ρ
(L.7)
On the other hand the parameter η3 can also be written as the product of
other two dimensionless parameters
η3 =
µU
2
R ρs g
=
νU ρs
= η4 η5 ,
R2 g ρ
(L.8)
Principles of Fluid Dynamics (www.fluiddynamics.it)
where
η4 =
νU
2
R g
and
η5 =
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ρs
.
ρ
The dimensionless parameters η3 , η4 and η5 are very different from the parameters η1 and η2 strictly related to the physics of the problem. Not only, but
from the parameters defined in the solution is not possible to go back to the dimensionless parameters appearing in the equations governing the problem. Any
information about the weight of the ball and the drag force is completely lost.
However, if a problem is defined by certain equations, the dimensionless parameters that are really meaningful are the parameters appearing in the equations, representing the basic physical processes. Even if different sets of dimensionless parameters can be mathematical equivalent, however, some of them are
physically more meaningful than others.
This confirms that if two quantities have the same unit of measurement, but
have meanings and play different roles within the laws governing the phenomenon,
then, they must be made different in some way. For example the density of the
fluid contributes to the drag force, while the density of the sphere contributes to
its weight, so that they cannot be treated in the same way. We must consider
such quantities as having different dimensions.
But when the density of the fluid is also used to evaluate the Archimedes’
correction, then it must be considered as a quantity of the same dimension as
the density of the sphere. A solution incorporating the effects of the Archimedes’
law would see the density of the water in two completely different roles: in other
words the same quantity might have different dimensions even within the frame
of the same problem.
Problem L.1 Include the Archimedes’ force in the solution of the problem above.
Answer. The drag force must be in equilibrium with the vertical gravitational force, that
in this case reads
4
P = πR3 (ρs − ρ)g.
3
Thus the final solution is
U =
2 R2 g ρs − ρ
.
9 ν
ρ
If we consider ρ as having two different dimensions in the numerator and denominator, the last fraction is no longer a pure number. Thus a decomposition of the kind
(L.8) is no longer possible, and the only possible decomposition is of the kind (L.6).
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L.5
Franco Mattioli (University of Bologna)
Transient velocity of a sinking ball
The problems studied so far are independent of time. Now we will consider
a problem containing the time as well, that is, the transient motion of the ball
before it reaches its limit velocity, within the hypothesis of vanishing initial velocity.
The equation of motion is
m
where
du
= − d + P,
dt
(L.9)
4
m = πR3ρs
3
is the mass of the ball, P = mg is its weight, u is the velocity of descent and d
is the instantaneous drag force
d = 6πρνRu.
For convenience, in (L.9) the motion is oriented downwards.
If (L.9) is divided by P , we obtain
1 du
g dt
=
9 ν
ρ
2
2 R g ρs
u − 1.
(L.10)
The previous problem suggests to introduce the terminal velocity U definite by
(L.7) to write
u = U u0 ,
where u0 is the velocity made dimensionless with respect to its maximum value.
If we furthermore set
t = T t0 ,
the first term of the equation becomes
U du0
.
gT dt0
By choosing
T =
U
2 R2 ρs
=
,
g
9 ν ρ
Principles of Fluid Dynamics (www.fluiddynamics.it)
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(L.10) can be rewritten in the dimensionless form
∂u0
= −u0 + 1.
∂t0
(L.11)
With the formalism adopted previously, (L.11) becomes
∂η2
= −η2 + 1.
∂η1
The solution of (L.11) is
u0 = 1 − e−t
0
,
from which the meaning of T as the time in which the gap between the velocity
and the terminal velocity is reduced by a factor e can be deduced.
In the present formulation time and velocity have been nondimensionalized on
the basis of quantities defined in the problem. Furthermore, in a time-dependent
motion, the two dimensionless parameters are related, because from a scale of
velocity and a scale of time a scale of space can be deduced. Thus, a phenomenon
developing on a greater space scale must evolve less rapidly.
Problem L.2 Determine the space scale L implicit in the present problem.
Answer.
U2
4 R4 g
L = UT =
=
g
81 ν 2
ρs
ρ
!2
.
Comment. The space scale L is the distance covered in the time interval T at the velocity
U . It provides an estimate of the order of magnitude of the distance at which the terminal
velocity will be reached.
L.6
Turbulent flow around a ball
Let us consider now the case in which the motion is no longer laminar, but
fully turbulent. As soon as turbulence develops, in the equations for the momentum of the fluid we have to take into account the terms of total derivative of the
velocity
∂u
1
+ u · ∇u = − ∇p + ν∇2 u.
(L.12)
ρ
∂t
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Franco Mattioli (University of Bologna)
Let us nondimensionalize the spatial coordinates as in (L.1). A time scale can
be derived from the space and velocity scales
t=
R 0
t.
U
Introducing these changes of variables, (L.12) becomes
0
νU 2
1
U 2 ∂u
U2 0
+
u · ∇0 u0 = − ∇0 p + 2 ∇0 u0 ,
0
R ∂t
R
ρR
R
with the same definition of ∇0 and ∇02 given in (L.2). Let us divide the equation
by U 2 /R. We obtain
∂u0
1 02 0
0
0 0
0 00
∇ u,
0 + u · ∇ u = −∇ p +
∂t
Re
(L.13)
where Re = U R/ν is the Reynolds number and
p = ρU 2 p00 .
Note that the dimensionless pressure p00 is different from the dimensionless pressure p0 = p00 Re defined in (L.3) for the creeping flow.
The drag force has the dimensions of a pressure multiplied by the surface of
the sphere
D = [p][R2 ] = [ρU 2 ][R2 ] = [ρU 2 R2 ],
so that
D
ρU 2 R2
= f (Re) ,
since the only dimensionless parameter present in the equations is the Reynolds
number.
The solution (L.4) for the creeping flow can be recovered by imposing
f (Re) = c
1
.
Re
There is no particular physical meaning in this relation. It is only matter of mathematical relations connecting the different dimensionless parameters adopted to describe the
creeping and the turbulent flows.
Principles of Fluid Dynamics (www.fluiddynamics.it)
xcvii
For very high Re numbers the last term of (L.13) vanishes, so that the dimensionless drag force cannot depend on the Reynolds number, which no longer
appears in the governing equations. Therefore the drag force must be constant
D = cD ρU 2 R2
(L.14)
where cD is a drag coefficient that can only be determined experimentally (see
(J.9).
The drag force is determined by the Reynolds stresses, whose magnitude depends
mainly on the largest fluctuations, which are of order R. The energy apparently dissipated by the largest fluctuations, is indeed dissipated by viscosity. The greater is the
energy to be dissipated, the smaller will be the scales at which viscosity becomes effective
in dissipating energy. Therefore, the magnitude of ν plays no role in the determination
of the drag force.
In order to have a good application of the drag force (L.14) the surface of the ball
should be rough, in such a way to facilitate the transition to a fully turbulent motion.
For intermediate values of the Reynolds number both viscosity and inertia
play some role, and we have a complex law for the drag force, which is not even
monotone, as one might imagine at a first sight.
The flow in fact does not pass directly from laminar to fully turbulent, but fields
(typically vortices) of various structure and behavior are generated.
L.7
Still on the meaning of dimension
We have seen that if we have at our disposal the equations of motion, we can
nondimensionalize them in such a way to reach easily, without the possibility of
errors, the correct nondimensionalization of the problem. We have also seen that
the parameters present in the solution can be different from the dimensionless
parameters present in the equations. Furthermore, from the latter, the former
cannot be easily deduced, unless the same physical quantity is considered has
having different dimensions when appearing in different physical roles.
This results is a step further with respect to the fact that two quantities can
have the same unit of measurement but different dimensions. The dimension is
related not only to the definition of a certain physical quantity, but also to the
role that the quantity plays in the given problem.
This justifies certain uncertainties of dimensional analysis and strongly suggests that a dimensional analysis must be always carried out on the governing
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Franco Mattioli (University of Bologna)
equations, when available. In many cases, when certain arguments are used, indeed, such arguments arise from the structure of the governing equations, even
when no explicit reference to them is made.
This happens, for example, when among the various dimensional parameters,
we introduce some that refer to the fundamental properties of the matter, such as
the density or the viscosity. In fact, we know the meaning of such parameters from
the structure of the fundamental laws of Physics, without which their definition
would not be possible.
L.8
About the applications of dimensional analysis
The dimensional analysis allows us to obtain useful conclusion about the
structure of the solution of a problem, even without solving the equations of
motion or when the equations of motion simply do not exist.
Probably, the main application of the dimensional analysis in fluid dynamics
refers to turbulent or chaotic motions, where satisfactory governing equations
are not known. In this case, we have also the certitude that such problems will
remain impossible to solve with alternative approaches for a long time to come.
The presence in a fluid of two very different regimes, the laminar and the
turbulent regimes, makes furthermore of fundamental importance the knowledge
of the magnitude of certain quantities that can help in the detection of the regime.
However, the dimensional analysis is not able to define accurately the separation
between the two kind of motion, that are known only via experimental data. It
can, however, provide asymptotic solutions.