3Dynamics
IB Fluid Dynamics
3.3 Reynolds number
As we mentioned, the Navier-Stokes equation is very difficult to solve. So we
want to find some approximations to the equation. We would like to know if we
can ignore some terms. For example, if we can neglect the viscous term, then we
are left with a first-order equation, not a second-order one.
To do so, we look at the balance of terms, and see if some terms dominate
the others. This is done via Reynolds number.
We suppose the flow has a characteristic speed
U
and an extrinsic length
scale
L
, externally imposed by geometry. For example, if we look at the flow
between two planes, the characteristic speed can be the maximum (or average)
speed of the fluid, and a sensible length scale would be the length between the
planes.
Next, we have to define the time scale
T
=
L/U
. Finally, we suppose pressure
differences have characteristic magnitude
P
. We are concerned with differences
since it is pressure differences that drive the flow.
We are going to take the Navier-Stokes equation, and look at the scales of
the terms. Dividing by ρ, we get
∂u
∂t
+ u · ∇u = −
1
ρ
∇p + ν∇
2
u,
where again ν =
µ
ρ
. We are going to estimate the size of these terms. We get
U
(L/U)
U ·
U
L
1
ρ
P
L
ν
U
L
2
.
Dividing by U
2
/L, we get
1 1
P
ρU
2
ν
UL
.
Definition (Reynolds number). The Reynolds number is
Re =
UL
ν
,
which is a dimensionless number. This is a measure of the magnitude of the
inertia to viscous terms.
So if
Re
is very large, then the viscous term is small, and we can probably
neglect it. For example, for an aircraft, we have
U ∼
10
4
,
L ∼
10 and
ν ∼
10
−5
.
So the Reynolds number is large, and we can ignore the viscous term. On the
other hand, if we have a small slow-moving object in viscous fluid, then
Re
will
be small, and the viscous term is significant.
Note that the pressure always scales to balance the dominant terms in the
equation, so as to impose incompressibility, i.e.
∇ · u
= 0. So we don’t have to
care about its scale.
In practice, it is the Reynolds number, and not the factors
U, L, ν
individually,
that determines the behaviour of a flow. For example, even though lava is very
very viscous, on a large scale, the flow of lava is just like the flow of water in a
river, since they have comparable Reynolds number.
Definition
(Dynamic similarity)
.
Flows with the same geometry and equal
Reynolds numbers are said to be dynamically similar.
When Re 1, the inertia terms are negligible, and we now have
P ∼
ρνU
L
=
µU
L
.
So the pressure balances the sheer stress. We can approximate the Navier-Stokes
equation by dropping the term on the left hand side, and write
0 = −∇p + µ∇
2
u,
with the incompressibility condition
∇ · u = 0.
These are known as Stokes equations. This is now a linear equation, with four
equations and four unknowns (three components of
u
and one component of
pressure). We find
u ∝ ∇p,
and so the velocity is proportional to the pressure gradient.
When
Re
1, the viscous terms are negligible on extrinsic length scale.
Then the pressure scales on the momentum flux,
P ∼ ρU
2
,
and on extrinsic scales, we can approximate Navier-Stokes equations by the
Euler equations
ρ
Du
Dt
= −∇p
∇ · u = 0.
In this case, the acceleration is proportional to the pressure gradient.
Why do we keep saying “extrinsic scale”? Note that when we make this
approximation, the order of the differential equation drops by 1. So we can no
longer satisfy all boundary conditions. The boundary condition we have to give
up is the no-slip condition. So when we make this approximation, we will have
to allow the fluid at the boundary to have non-zero velocity.
So when is this approximation valid? It is obviously wrong when we are at
the boundary. If the velocity gradient near the boundary is relatively large, then
we quickly get significant non-zero velocity when we move away from boundary,
and hence obtain the “correct” answer. So we get problems only at the length
scale where the viscous and inertia effects are comparable, i.e. at the intrinsic
length scale.
Since the intrinsic length scale
δ
is the scale at which the viscous and inertia
terms are comparable, we need
U
2
∼
νU
δ
2
.
So we get
δ =
ν
U
.
Alternatively, we have
δ
L
=
ν
UL
=
1
Re
.
Thus, for large Reynolds number, the intrinsic length scale, which is the scale
in which the viscous and boundary effects matter, is small, and we can ignore
them.
For much of the rest of this course, we will ignore viscosity, and consider
inviscid flow.