4.3 Fluid Dynamics
99
R e = ρv f L/η ,
(4.44)
a quantity to which we referred earlier when describing the drag on a body in a
moving fluid (see Sect. 3.3.5). We may think of the Reynolds number as the ratio of
a characteristic convective inertial momentum to the impulse on the body due to the
viscous force.
If we measure all distances relative to L, all times relative to L/v f , define a
unitless pressure by p = p/(ρv 2
f ), and a unitless gravitational potential by
g =
g /v 2
f , then the Navier–Stokes equation (4.43) will become
∂v
∂t + v
· ∇
v
= −∇
p
+
1
R e
∇
2 v
− ∇
g .
(4.45)
This is marvelous, because we can study the flow, say in a wind tunnel, for one
set of values of L and v f , and then re-scale to find a family of solutions for differing
values of L and v f , as long as we keep the same Reynolds number and boundary
conditions. This can be done for model airplanes and model birds, with re-scaling
to give solutions for real airplanes and real birds.
Another interesting dimensionless quantity for a fluid is the ‘Rayleigh number’
which, for given boundary conditions, determines if the fluid will have convective
instability due to the buoyancy of heated fluid. It is given by
R a =
ρgααT V
ηκ D
,
(4.46)
where V is the volume of the fluid element being buoyed up, ρ its density, g the
local gravitational acceleration, α the thermal expansivity of the fluid, T the
temperature decrease from bottom to top of the fluid element, η is the fluid viscosity,
and κ D the thermal diffusivity. For example, if you sit in a room of initially still air,
a convective plume of heated air will rise above your head. As the Rayleigh number
increases, the plume can divide into smaller convective zones, show flapping zones
and puffing behavior, and finally chaotic turbulence.
4.3.8 Streamline Versus Turbulent Flow
Streamline flow is characterized by being steady, with adjacent fluid element
following almost parallel paths. Such flow is also referred to as ‘laminar flow’, i.e.
flow in layers.
Fluid flow is ‘turbulent’ if it is unsteady and irregular. A threshold size of the
Reynolds number is observed to characterize fluid instability at the onset of a
transition of fluid flow from streamline to turbulent. Figure 4.9 shows how a fluid
such as water might flow past a ball (fluid enters from left). The top drawing depicts
streamline flow (R e < 1), for which Stokes’ law applies; the center drawing depicts
99
R e = ρv f L/η ,
(4.44)
a quantity to which we referred earlier when describing the drag on a body in a
moving fluid (see Sect. 3.3.5). We may think of the Reynolds number as the ratio of
a characteristic convective inertial momentum to the impulse on the body due to the
viscous force.
If we measure all distances relative to L, all times relative to L/v f , define a
unitless pressure by p = p/(ρv 2
f ), and a unitless gravitational potential by
g =
g /v 2
f , then the Navier–Stokes equation (4.43) will become
∂v
∂t + v
· ∇
v
= −∇
p
+
1
R e
∇
2 v
− ∇
g .
(4.45)
This is marvelous, because we can study the flow, say in a wind tunnel, for one
set of values of L and v f , and then re-scale to find a family of solutions for differing
values of L and v f , as long as we keep the same Reynolds number and boundary
conditions. This can be done for model airplanes and model birds, with re-scaling
to give solutions for real airplanes and real birds.
Another interesting dimensionless quantity for a fluid is the ‘Rayleigh number’
which, for given boundary conditions, determines if the fluid will have convective
instability due to the buoyancy of heated fluid. It is given by
R a =
ρgααT V
ηκ D
,
(4.46)
where V is the volume of the fluid element being buoyed up, ρ its density, g the
local gravitational acceleration, α the thermal expansivity of the fluid, T the
temperature decrease from bottom to top of the fluid element, η is the fluid viscosity,
and κ D the thermal diffusivity. For example, if you sit in a room of initially still air,
a convective plume of heated air will rise above your head. As the Rayleigh number
increases, the plume can divide into smaller convective zones, show flapping zones
and puffing behavior, and finally chaotic turbulence.
4.3.8 Streamline Versus Turbulent Flow
Streamline flow is characterized by being steady, with adjacent fluid element
following almost parallel paths. Such flow is also referred to as ‘laminar flow’, i.e.
flow in layers.
Fluid flow is ‘turbulent’ if it is unsteady and irregular. A threshold size of the
Reynolds number is observed to characterize fluid instability at the onset of a
transition of fluid flow from streamline to turbulent. Figure 4.9 shows how a fluid
such as water might flow past a ball (fluid enters from left). The top drawing depicts
streamline flow (R e < 1), for which Stokes’ law applies; the center drawing depicts