Consider the flow of a fluid over a flat plate (Fig. 2.1.1).
Figure 2.1.1 Boundary Layer Development along a Flat Plate
Beginning at the leading edge of the plate, a region develops where the influence of viscous forces is felt. These viscous forces are described in terms of a shear stress, τ, between the fluid layers. If this stress is assumed to be pro-portional to the normal velocity gradient, the defining equation for viscosi-ty, known as Newton’s equation of viscosiviscosi-ty, is
(2.1.1)
The constant of proportionality, µ, is called the dynamic viscosity. The values of µ for some fluids are given in appendix A.
dv τ = –µ dy
The velocity or hydrodynamic boundary layer is the region of flow that develops from the leading edge of the plate in which the effects of viscosity are observed. The y-position where the boundary layer ends is arbitrarily chosen at a point where the velocity becomes 99% of the free stream value.
The boundary layer thickness, δ, is defined as the distance between this point and the plate.
Initially, the boundary layer development is laminar. At some critical distance from the leading edge, small disturbances in the flow begin to become amplified and a transition process takes place until the flow becomes turbulent. This depends on the flow field and fluid properties.
The physical mechanism of viscosity is one of momentum exchange. In the laminar portion of the boundary layer, molecules move from one lamina to another and carry momentum corresponding to the velocity of the flow.
There is a net momentum transport from regions of high velocity to regions of low velocity, which creates a force in the direction of flow. This force may be expressed in terms of the viscous shear stress as given by Equation 2.1.1.
The rate at which the momentum transfer takes place is dependent on the rate at which the molecules move across the fluid layers. In a gas, the molecules would move about with some average speed proportional to the square root of the absolute temperature since we identify temperature with the mean kinetic energy of a molecule in the kinetic theory of gases. The faster the molecules move, the more momentum they will transport. Hence we should expect the viscosity of a gas to be approximately proportional to the square root of temperature, and this expectation is corroborated fairly well by experiment.
The laminar velocity profile is approximately parabolic in shape. The transition from laminar to turbulent flow occurs typically when
where
v∞ = free stream velocity
x = distance from the leading edge of the plate ν = µ/ρ, the kinematic viscosity of the fluid
qv∞x l
v∞x
= m ≥3.2x105
This particular dimensionless group or ratio of inertial force to viscous force is called the Reynolds number after the British scientist-engineer who first did extensive research on flow in the late 1800s.
(2.1.2)
Although the critical Reynolds number for transition on a flat plate is usually taken as 3.2 x 105for most analytical purposes, the critical value in a practical situation is strongly dependent on the surface roughness conditions and the turbulence level of the free stream. The normal range for the beginning of transition is between 3.2 x 105and 106. With very large disturbances present in the flow, transition may begin with Reynolds numbers as low as 105. For flows which are very free from fluctuations, it may not start until Rex = 2 x 106 or more. In reality, the transition process covers a range of Reynolds numbers. Completed transition and fully developed turbulent flow usually is observed at Reynolds numbers twice the value at which transition began.
A qualitative picture of the turbulent flow process may be obtained by imagining macroscopic chunks of fluid transporting momentum instead of microscopic transport on the basis of individual molecules. The turbulent boundary layer is more complex than the laminar boundary layer because the nature of the flow in the former changes with distance from the plate surface.
The zone adjacent to the wall is a layer of fluid, which, because of the stabilizing effect of the wall, remains laminar even though most of the flow in the boundary layer is turbulent. This very thin layer is called the laminar sublayer, and the velocity distribution in this layer is related to the shear stress and viscosity using Newton’s viscosity law.
The flow zone outside the laminar sublayer is turbulent. The turbulence alters the flow regime so much that the shear stress, as given by τ = - µ dv/dy, is not significant. The mixing action of turbulence causes small fluid masses to be swept back and forth in a direction transverse to the mean flow direction.
As a small mass of fluid is swept from a low-velocity zone next to the sublayer into a relatively high-velocity zone farther out in the stream, the mass has a retarding effect on the high-velocity stream. This mass of fluid, through an exchange of momentum, creates the effect of a retarding shear stress applied to a high-velocity stream. A small mass of fluid originates farther out in the boundary layer in a high-velocity flow zone and is swept into a region of
ρv∞x Rex = µ
relatively low velocity. This has an effect on the low-velocity fluid much like shear stress augmenting the flow velocity. In other words, the mass of fluid with relatively higher momentum will tend to accelerate the lower velocity fluid in the region into which it moves. Although the process described previously is a momentum-exchange phenomenon, it has the same effect as a shear stress applied to the fluid. In turbulent flow, these stresses are termed apparent shear stresses or Reynolds stresses. The turbulent velocity profile has a nearly linear portion in the sublayer and a relatively flat profile outside this region.
Consider the flow in a tube shown in Figure 2.1.2. A boundary layer develops at the entrance. Eventually the boundary layer fills the entire tube, and the flow is said to be fully developed. If the flow is laminar, a parabolic velocity profile is experienced as illustrated in Figure 2.1.2a. When the flow is turbulent, a somewhat blunter profile is observed (Fig. 2.1.2b). In a tube, the Reynolds number based on the mean fluid velocity and the tube diameter is again used as a criterion for laminar and turbulent flow. For Red= ρvd/µ ≤ 2300, the flow is usually observed to be laminar, whereas for Red≥ 10,000, it is turbulent.
Fig. 2.1.2 Velocity Profiles in a Tube: (a) Laminar Flow (b) Turbulent Flow
Again, a range of Reynolds numbers for transition may be observed depending on the roughness of the pipe and smoothness of the flow. The generally accepted range for transition, also referred to as the critical region, is 2000 < Red < 4000. Laminar flow has been maintained up to Reynolds numbers of 25,000 in carefully controlled laboratory conditions.
The mass flow rate or continuity relationship for one-dimensional flow in a tube is
(2.1.3)
where
m = mass rate of flow v = mean velocity
A = cross-sectional area of the tube
The mass flux or mass velocity is defined as
(2.1.4)
so the Reynolds number may be written as
(2.1.5)
Similar flow patterns are observed in ducts that do not have a circular cross section. In those cases, it is convenient to define the following equivalent or hydraulic diameter for calculating the Reynolds number:
(2.1.6)
This particular grouping of terms is used because it yields the value of the physical diameter when applied to a circular cross section.
de = 4 x cross-sectional flow area wetted perimeter
Red = Gd/µ G = m/A = ρv
m = ρvA