6. SIMILITUDE AND DIMENSIONAL ANALYSIS
It is usually impossible to determine all the essential facts for a given fluid flow by pure theory, and hence dependence must often be placed upon experimental investigations. The number of tests to be made can be greatly reduced by a systematic program based on dimensional analysis and specifically on the laws of similitude or similarity, which permit the application of certain relations by which test data can be applied to other cases.
Thus the similarity laws enable us to make experiments with a convenient fluid such as water or air, for example, and then apply the results to a fluid which is less convenient to work with, such as gas, steam or oil. Also, valuable results can be obtained as a minimum cost by tests made with small-scale models of the full-sized apparatus. The laws of similitude make it possible to determine the performance of the prototype, which means the full-size device, from tests made with the model. It is not necessary that the same fluid be used for the model and its prototype. Neither the model necessary smaller than its prototype.
6.1. Geometric similarity
One of the desirable features in model studies is that there be geometric similarity, which means that the model and its prototype be identical in shape but differ only in size. The important consideration is that the flow patterns be geometrically similar. If the scale ratio is denoted by Lr, which means the ratio of the linear dimensions of the prototype to corresponding dimensions in the model, it follows that areas vary as L2r and volumes as L3r. Complete geometric similarity is not always easy to attain. Thus the surface roughness of a small model my not be reduced in proportion unless it is possible to make its surface very much smoother than that of the prototype.
6.2. Kinematic similarity
Kinematic similarity implies geometric similarity and in addition it implies that the ratio of the velocities at all corresponding points in the flow is the same. If subscripts p and m denote prototype and model, respectively, the velocity ratio v, is
m p
r v
v = v (6.1)
and its value in terms of Lr will be determined by dynamic considerations.
As time T is dimensionally L / v, the time scale is
r r r
v
T = L (6.2)
and in a similar manner the acceleration scale is
r r r
r
r L
v T
a = L2 = 2 (6.3)
6.3. Dynamic similarity
If two systems are dynamically similar, corresponding forces must be in the same ratio in the two. Forces that may act on a fluid element include those due to gravity FG, pressure FP, viscosity FV, and elasticity FE. Also, if the element of fluid is at a liquid-gas interface, there are forces due to surface tension FT. If the summation of forces on a fluid element does not add up to zero, the element will accelerate in accordance to Newton’s second law. Such an unbalanced force
system can be transformed into a balanced system by adding an inertia force FI that is equal and opposite to the resultant R of the acting forces. Thus generally,
R F F F F F
F = G + P + V + E + T =
∑
(6.4)and
R
FI =− (6.5)
Thus
=0 + + + +
+ P V E T I
G F F F F F
F (6.6)
These forces may be expressed in simplest terms as follows:
Gravity FG =mg=ρL3g Pressure FP =
( )
∆p A=( )
∆p L2 ViscosityL v L L
A v dy
FV du = ⋅ ⋅
=
=η η 2 η
Elasticity FE =EvA=EvL2 Surface tension FT =σL
Inertia 2 2 2
4 2
3 v L
T L T
L L ma
FI = =ρ =ρ =ρ
In many flow problems some of these forces are either not present or insignificant. Consider, for example, the case where the forces acting on a fluid system are FG, FP, FV, and FI. Then dynamic similarity will be achieved if
FIm
F F F F F F
F Ip
Vm Vp Pm Pp Gm
Gp = = = (6.7)
where subscripts p and m refer to prototype and model, respectively. These relations can be expressed as
G m I G p
I
F F F
F
=
,
P m I P p
I
F F F
F
=
,
V m I V p
I
F F F
F
=
(6.8)
Each quantity is dimensionless and the significance of the dimensionless ratios is discussed below.
Reynolds number
In the flow of a fluid through a completely filled pipe, gravity does not affect the flow pattern. It is also obvious that capillarity is of no practical importance, and hence the significant forces are inertia and fluid friction due to viscosity.
Considering the ratio of inertia forces to viscous forces, the parameter obtained is called the Reynolds number, or “Re”. The ratio of the two forces is
ν η
ρ η
ρ Lv Lv
Lv v L F F
V
I = = =
= 2 2
Re (6.9)
For any consistent system of units, Re is a dimensionless number. The linear dimension L may be any length that is significant in the flow pattern. Thus, for a pipe completely filled, it might be either the diameter or the radius, and the numerical value of Re will vary accordingly. General usage prescribes L as the pipe diameter.
Figure 6.1 depicts two simple illustrations. The first involve a fluid that flows with velocity U0 through a channel of width w that makes a sudden right turn. During the turn time τ0 ~wU0, a fluid element rounding the corner loses momentum density ρU0 by exerting an inertial centrifugal density fi ~ρU0 τ0 =ρU02 w. The other involves a fluid flowing in a channel that contracts over a length l. By mass conservation, the velocity increases as u~U0
(
1+z/l)
, causing a fluid element to gain momentum at a ratel U dz U du dt
fi ~ ρdu =ρ 0 ~ ρ 02 . (6.10)
The force exerted on the rest of the fluid is equal and opposite to the force required to accelerate each fluid element. Thus the force on the fluid due to a curved streamline points outwards centrifugally, and the inertial force in an expansion or contraction points towards the wide end of the channel, regardless of the flow direction. In both cases, the magnitude of inertial and viscous force densities must be compared. Viscous force densities results from gradients in viscous stress, and thus scale as fv ~ηU0 L20, where L0 is a typical length scale. The ratio of these two force densities is just the Reynolds number
Re
0
0 =
= η
ρU L f
f
v
i (6.11)
Figure 6.1. Inertial forces exerted by accelerating fluid elements. (a) A small fluid element roundng a corner. (b) A fluid element flowing through a contraction.
Froude number
Considering inertia and gravity forces alone, a ratio is obtained called a Froude number, or “Fr”.
The ratio of inertia forces to gravity forces is
gL v gL
v
L 2
3 2
2 =
ρ
ρ (6.12)
Although this is sometimes defined as the Froude number, it is more common to use the square root so as to have v in the first power, as in the Reynolds number. Thus a Froude number is
gL
= v
Fr (6.13)
System involving gravity and inertia forces are the flow of water in open channels, the flow over a spillway, or the flow of a stream from an orifice.
For the computation of Fr, the length L must be some linear dimension that is significant in the flow pattern. For an open channel, for example, it is taken as the depth of flow.
The velocity ratio is
1
r m
p r
L v
v = v = (for same Fr) (6.14)
while the ratio of time for prototype to model is
1
r m
p r
L T
T =T = (for same Fr) (6.15)
and ar = 1.
Since the velocity varies as and the cross section area as , it follows that
1
2 / 5 r m
p r
L Q
Q = Q = (for same Fr) (6.16)
Mach number
Where compressibility is important, it is necessary to consider the ratio of the fluid velocity to that of a sound wave in the same medium. This ratio, called the Mach number, or “Ma”, is
c
= v
Ma (6.17)
where c is the acoustic velocity in the medium in question. If Ma is less than 1, the flow is called subsonic; if it is equal to 1, the flow is sonic; if it is greater than 1, the flow is called supersonic;
and for extremely high values of Ma the flow is called hypersonic.
The ratio of inertia forces to elastic forces
v
v E
v L E
L
v 2
2 2
2 ρ
ρ = (6.18)
is called the Cauchy number. The celerity c of an acoustic wave is given by
ρv
c= E (6.19)
and, therefore, the Cauchy number is the square of the Mach number Ma2
Ca= (6.20)
Weber number
In a few cases of flow, surface tension may be important, but normally it is negligible. The ratio of inertia forces to surface tension forces is ρv2L2
( )
σL , the square root of which is known as the Weber number:L v ρ
= σ
We (6.21)
Euler number
A dimensionless quantity related to the ratio of the inertia forces to the pressure forces is known as the Euler number. It is expressed as
γ ρ
p g v p
v
= ∆
= ∆
2
Eu 2 (6.22)
If only pressure and inertia influence the flow, the Euler number will remain constant.
Weissenberg and Deborah numbers
Thus far, we have considered the influence of inertia, pressure, viscosity, surface tension, compressibility and gravity on fluid flows. Dissolved polymers, for example, add an elastic component to the fluid that further enriches flow behavior. We provide here a simple picture of polymer dynamics to illustrate the basic physics at hand. The simplest model system treats a polymer as a dumbbell with two beads each of hydrodynamic resistance ξ connected by an entropic spring of stiffness kH. The spring constant kH can be obtained for a freely jointed chain with N steps of length b as kH = 3kBT/Nb2. For simplicity, we confine the polymer to the one-dimensional line along the center of a contracting channel, as in Figure 6.2. Due to the contraction, the local (extensional) flow felt by the polymer increases as it moves down the channel, roughly like
z e u
uz = 0 +& . The bead separation Rp changes due to three physical effects: (i) Brownian motion
drives beads apart with average speed RpR&p ~D~kBT/ξ [the diffusivity D~kBT/(6πηa) for spherical solute molecules of size a], (ii) the spring pulls the beads together with velocity
ξ /
~ H p
p k R
R& − , and (iii) the extensional flow advects the forward bead more quickly than the rear
bead, driving them apart with velocity R&p ~e&Rp. Each effect has its own characteristic time scale:
D
D =R02/
τ for the beads to diffusively explore a length scale R0, τp =ξ/kH for the polymer spring to relax, and τe=e&−1 is a time scale associated with extensional flow.
While this example helps in developing physical insight and intuition, its applicability is generally limited. It assumes polymers do not interact, which requires the solution to be dilute. The solution stress is assumed to be dominant by the viscous solvent rather than the polymers, constituting a so-called Boger fluid. Polymer deformations are also assumed to be small since we
have ignored the nonlinear effects for large deformations. Despite these simplifications, several important phenomena are illustrated by the above simple example.
Figure 6.2. The effect of a non-uniform flow on a model polymer, represented as two beads separated by a distance R on a spring. (a) The center of mass of the polymer moves with velocity UCM, whereas the two
beads experience relative motion UR~e&Rp~kHRp/ξ due to external flow gradients and spring forces.
Brownian forces balance spring forces to give a steady-state polymer size. When Wi ≥ O(1), the extensional flow overwhelms the spring, and the polymer unravels via a coil-stretch transition. (b) As the beads are
pulled toward the spring, each exerts a force back on the fluid, resulting in a force-dipole flow.
Weissenberg number
In equilibrium e&=0, spring forces balance Brownian forces to give a characteristic polymer size R0 ~
(
kBT kH)
1/2 ~( )
Nb2 1/2. By contrast, an extensional flow acts to drive the beads apart (like a negative spring), and alters the steady polymer size via(
0)
1/22 / 1
Wi
~ 1
~ )
( −
−
R e
k T e k
R
H
p & B &ξ . (6.23)
Here we have introduced the Weissenberg number
γ τ
τpe& or p&
Wi= (6.24)
which relates the polymer relaxation time to the flow deformation time, either inverse extension rate
e&or shear rate γ&. When Wi is small, the polymer relaxes before the flow deforms it significantly,
and perturbations to equilibrium are small. As Wi approaches 1, the polymer does not have time to relax and is deformed significantly.
Deborah number
Another relevant time scale τflow characteristic of the flow geometry may also exist. For example, a channel that contracts over a length L0 introduces a geometric time scale τflow =L0/U0 required for a polymer to transverse it. The flow time scale τflow can be long or short compared with the polymer relaxation time τp, resulting in a dimensionless ratio known as the Deborah number
flow p
τ
= τ
De . (6.25)
Note that the usage of De and Wi can vary. Some references use Wi exclusively to describe shear flows and use De for the general case, whereas others use Wi for local flow time scales due to a local shear and De for global flow time scales due to a residence time in flow.
Elasticity number
As the flow velocity U0 increases, elastic effects become stronger and De and Wi increase.
However, the Reynolds number Re increases in the same way, so that inertial effects become more important as well. The elasticity number
El = De / Re = 2 h
p
ρ η
τ (6.26)
where h is the shortest dimension setting shear rate, expresses the relative importance of elastic to inertial effects. Significantly, El depends only on the geometry and material properties of the fluid, and is independent of flow rate.
Knudsen number
We have implicitly focused on liquids thus far, and have not explicitly discussed gas flows. The molecular-level distinction between liquids and gases can have important ramifications for fluid flows. While liquid molecules are in constant collision, gas molecules move ballistically and collide only rarely. Using the kinetic theory, one can calculate the mean free path between collisions to be
2
~ 1
f na
λ , (6.27)
where n is the number density of molecules with radius a. For example, an ideal gas at 1 bar and 25oC has a mean free path λf ~70 nm that increases at lower pressures or higher temperatures.
As the flow geometry get smaller, the mean free path occupies an increasingly significant portion of the flow, and thus plays an increasingly important role. The Knudsen number
L λf
=
Kn (6.28)
expresses the ratio of the mean free path (the length scale on which molecules matter) to a macroscopic length scale L. The latter is typically a length scale representative of the device, but could also be given by the length scale for temperature, pressure, or density gradients.
Noncontinuum effects play an increasing role as Kn increases. Roughly speaking, molecules located farther than λf from a solid wall do not see the wall, whereas closer molecules can collide with the wall rather than other molecules. This implies that the fluid behaves like a continuum up to a distance λf from the wall, and influences the boundary conditions obeye by the fluid. Maxwell was the first who predicted the no-splip boundary condition, yielding instead the slip condition
0
0 dn
u =βdu (6.29)
where β is a slip length of order λf.
On the other hand, the density of a gas typically depends much more strongly on temperature and pressure than that of a liquid. Therefore, compressibility can play a much more important role in gas flows, particularly when significant differences in T or p exists. Both fluid density and mean free path λf are affected. Three distinct Kn regimes have been measured in pressure-driven flows: Kn << 1, where the gas behaves as a no-slip fluid, Kn ~ 1, where the gas
behaves as a continuum but slips at the boundaries, and Kn >> 1, where the continuum approximation breaks down completely.
Liquid molecules are in constant contact, resulting in significantly higher incompressibility and making the concept of λf less meaningful. As such, noncontinuum effects appear to play a role only when the fluids themselves are confined to molecular length scales.
A fundamental question that arises is whether fluid truly slips over a solid surface, invalidating the no-slip boundary condition, or whether the experiments reflect an apparent slip that arises from surface inhomogeneities or a complex interface with additional physics. Recent molecular dynamics simulations, whose physical ingredients are known, have indeed shown fluid to slip (albeit with much shorter slip length), particularly when the attraction between molecules is stronger than the wall/molecule attraction. Various explanations have been proposed to account for apparent slip effects. A perfectly slipping but rough surface obeys a macroscopic boundary condition with slip length of order the roughness length scale. From this standpoint, any macroscopic length (real or apparent) will be confined to the roughness scale. However, even nonwetting, molecularly smooth surfaces have exhibited large apparent slip lengths. A gas layer at the interface, which would lubricate and alter the fluid flow, has been invoked as an explanation.
The idea is that dissolved gas molecules are drawn out of solution to nucleate a gas layer at the solid-liquid interface and lower the surface energy.
The no-slip boundary condition never enjoyed solid theoretical background, but rather was accepted due to its apparent experimental success. An understanding of the apparent slip might allow surfaces to be designed specifically to slip, e.g. reduce hydrodynamic resistance. In this case, an analogous Knudsen number
L
= β
Kn (6.30)
will be significant in the description of such flows.
Bond number and capillary number
In some cases, surface tension and gravity may be important. The ratio of gravity forces to surface tension forces is ρv2L2
( )
σL , which is known as the Bond number:σ ρL2g
Bo= (6.31)
Consider now a long droplet of length L in a channel of radius w with an inhomogeneous surface:
hydrophobic (with σslL) for z < 0 and hydrophilic (with σ >slR σslL) for z > 0. Energetically, the droplet wants to move onto the hydrophilic surface, and moving with velocity U decreases the stored interfacial energy at a rate ~
( )
∆σ wU , where ∆σ =σslL −σslR. This energy is lost to viscous dissipation, which consumes a power η∫ (∂u ∂z)
2dV ~ηU2L when dissipation is dominated by
viscous shear in the bulk. Assuming the capillary energy released to be balanced by viscous
dissipation gives
L w U = =
∆ Cp
σ
η (6.32)
The capillary number arises naturally, because capillary stresses are balanced by viscous stresses. In this case, the droplet moves with velocity scale U ~w∆σ ηL.
Figure 6.3. Droplet motion due to a gradient in solid-liquid interfacial energy
Illustrative example. A submerged body is to move horizontally through a liquid with γ = 8340 N/m3 and η = 1.5×10-3 Ns/m2 at a velocity of 2.5 m/s. To study the characteristics of this motion, an enlarged model is tested in a liquid with γ = 9810 N/m3 and η = 10-3 Ns/m2. The model ratio is 8:1.
Determine the velocity at which this enlarged model should be pulled through the water to achieve dynamic similarity. If the drag force (F ∝ ρL2v2) on the model is 10000 N, predict the drag force on the prototype. Reynolds criterion must be satisfied.
The parameters of the prototype are:
γp = 8340 N/m3 , ηp = 1.5×10-3 Ns/m2, vp = 2.5 m/s .
The known parameters of the model are:
γm = 9810 N/m3 , ηm = 10-3 Ns/m2, Fm = 10000 N, and
8
=1
m p
L
L .
m m m m
p p p
pL v L
v
ρ η ρ
η = ⇒ 8 2.5 1.417m/s
10 9810 8340
10 5 .
1 1
3 1 3
=
×
× × ×
=
=
−
−
− − p p m m m p p
m v
L v L
η γ γ η
0043 . 5 0 . 2 8 9810
417 . 1 8340
2 2
2 2
2 2 2
× =
×
= ×
⋅
⋅
⋅
= ⋅
m m m
p p p m
p
v L
v L F
F ρ
ρ ⇒ Fp = 0.0043 × Fm = 42.67 N
6.4. Comments on models
In the use of models it is essential that the fluid velocity should not be used so low that laminar flow exists when the flow in the prototype is turbulent. Also, conditions in the model should not be such that surface tension is important if such conditions do not exist in the prototype.
When modeling a subsonic flow of a body in a wind tunnel, it is commonly necessary to conduct the test under high pressure in order to satisfy the Reynolds criterion
p m
Dv
Dv
=
η ρ η
ρ (6.33)
without introducing compressibility effects. For example, suppose Lr = Dp / Dm = 20. If the viscosity and the density of the air were the same in the model and prototype, then to satisfy Reynolds’s criterion, vm = 20 × vp. For a body operating at low speed this would make the model Mach number much greater than one, and compressibility effects would invalidate the behaviour of the model. If, however, the test were conducted under a pressure of 20 bar with identical model and prototype temperatures ρm=20×ρp and ηm ≈ηp since the viscosity of air changes very little with pressure. In this case the model should be operated at a velocity equal to that of the prototype in order for the Reynolds number to be the same.
6.5. Dimensional analysis
Fluid mechanics problems may be approached by dimensional analysis, a mathematical technique making use of the study of dimensions. In dimensional analysis, from a general understanding of fluid phenomena, one first predicts the physical parameters that will influence the flow, and then, by grouping these parameters in dimensionless combinations, a better understanding of the flow phenomena is made possible.
To illustrate the steps in a dimensional-analysis problem, let us consider the drag force FD
exerted on a sphere as it moves through a viscous liquid. We must visualize the physical problem to consider what factors influence the drag force. Certainly, the size of the sphere must enter the problem; also, the velocity of the sphere must be important. The fluid properties involved are density ρ and the viscosity η. Thus we can write
(
D,v,ρ,η)
f
FD = (6.34)
Here D, the sphere diameter, is used to represent sphere size.
We want to determine how these variables are interrelated. Our approach is to satisfy dimensional homogeneity. That is, we want the dimensions on one side of the equation to correspond to those on the other. The preceding expression may be written as a power equation
d c b a
D CD v
F = ρ η (6.35)
where C is a dimensionless constant. Substituting the proper dimensions we get
d c b a
LT M L
M T L L T
ML
= 3
2 (6.36)
To satisfy dimensional homogeneity the exponents of each dimension must be identical on both sides of the equation. Thus
For M: 1 = c + d
For L: 1 = a + b – 3c – d For T: −2 = −b − d
Since we have three equations with four unknowns, we must express three of the unknowns in terms of the fourth. Solving for a, b, and c in terms of d, we get
a = 2 – d , b = 2 – d , c = 1 – d Thus
( ) ( ) d
D
v vD D C F
−
=
η
ρ 2 2 ρ (6.37)
It may be seen that the quantity vDρ η is a Reynolds number. Thus the original power equation can be expressed as
( )
Re D2v2 fFD = ′ ρ (6.38)
The results indicates that the drag on a sphere is equal to some coefficient times ρD2v2, where the coefficient is a function of the Reynolds number.
The foregoing approach to dimensional analysis is commonly referred to as the Rayleigh method. Another more generalized approach is through use of the Buckingham Π theorem. This theorem states that if there are n dimensional variables in a dimensionally homogeneous equation, described by m fundamental dimensions, they may be grouped in n-m dimensionless groups. Thus, in the preceding example, n = 5 and m = 3 (M, L, and T) and n-m = 2; these dimensionless groups were Re and FD/
(
ρD2v2)
.Applying the Π theorem to the preceding example, one would proceed as follows:
(
, , , ,)
=0′ F D v ρ η
f D (6.39)
where n = 5, m = 3, so n-m = 2. Thus we can write:
(
Π1,Π2)
=0φ (6.40)
The problem now is to find the Π’s by arranging the five parameters into two dimensionless groups.
Taking ρ, D, and v as the primary variables, the Π terms are:
1 1 1 1
1
d c b aD v η ρ
=
Π (6.41)
2 2 2
2 2
d D c b
a D v F
ρ
=
Π (6.42)
The values of the exponents are determined as before, noting that since the Π’s are dimensionless, they can be replaced with M0L0T0. Working with Π1,
1 1
1 1
3 0 0 0
d c
b a
LT M T L L L T M L
M
= (6.43)
thus
for M: 0 = a1 + d1
for L: 0 = −3a1 + b1 + c1 – d1
for T: 0 = −c1 – d1
Solving for a1, b1, and c1 in terms of d1,
a1 = −d1 , b1 = −d1 , c1 = −d1
Thus,
Re
1 1
1 1
1 1 =
=
=
Π − − −
d d
d d d
v Dv
D ρ
η η
ρ (6.44)
Working in a similar fashion with Π2, one gets
2 2 2
v D
FD
= ρ
Π (6.45)
Finally, φ
(
Π1,Π2)
=0 may be expressed as( )
Π= ′
Π φ or Π =φ′′
( )
Π (6.46)So
( )
Re2
2 φ
ρ = ′′
v D
FD
(6.47)
and FD =φ ′′
( )
Re ρD2v2 (6.48)It should be emphasized that dimensional analysis does not provide a complete solution to fluid problems. It provides a partial solution only. The success of dimensional analysis depends entirely on the ability of the individual using it to define the parameters that are applicable. If one omits an important variable, the results are incomplete and this may lead to incorrect conclusions.
On the other hand, if one includes a variable that is totally unrelated to the problem, an additional insignificant dimensionless group will result.
Illustrative example. Derive an expression for the flow rate q over the spillway shown in the accompanying figure. Assume that the sheet of water is relatively thick so that surface-tension effects may be neglected. Assume also that gravity effects predominate so strongly over viscosity that viscosity may be neglected.
Figure 6.4. Flow over a spillway
Under the assumed conditions the variables that effect q would be the head H, the acceleration of gravity g, and possibly the spillway height P. Thus
(
H g P)
f q= , ,
or f
(
q,H,g,P)
=0In this case n = 4, and m = 2. Hence, according to the theorem, there are n-m = 2 dimensionless groups, and
(
Π1,Π2)
=0 φUsing q and H as the basic variables,
1 1 1
1
c b
aH g
=q Π
2 2 2
2
c b
a H g
=q Π Working with Π1
1 1 1
2 0 3
0
c b a
T L L TL T L
L
=
for L: 0 = 2a1 + b1 + c1
for T: 0 = −a1 – 2c1
Hence
1
1 2
1a
c =− , 1 1 2 3a b =−
1 1
1 1
2 / 3 2 / 1 2
/ 1 2 / 3 1
a a
a a
H g g q
H
q
=
=
Π − −
Working with Π2
2 2
3 2
0
0 b c
a
L TL L
T L
L
=
for L: 0 = 2a2 + b2 + c2
for T: 0 = −a2 Hence
2 =0
a , c2 =−b2
2 2
0 2
2
b b
b
P P H
H
q
=
=
Π −
Finally, φ
(
Π1,Π2)
=0 can be written as( )
21 = Π
Π φ
= P H H
g
q 3/2 φ or
2 /
H3
P g
q H
=φ
Thus dimensional analysis indicates that the flow rate per unit length of spillway is proportional to gand to H3/2. The flow rate is also affected by the H/P ratio.
