Reynold’s no, Re
V = mean velocity of flow in pipe
D = Characteristic length of the geometry
μ = dynamic viscosity of the liquid (N – s /m2)
ν = Kinematic viscosity of the liquid (m2/s)
Ac = Cross – section area of the pipe
P = Perimeter of the pipe
Laminar flow in a pipe
Now, forces acting on the fluid element are:
(a). The pressure force, P×πR2 of face AB.
(b). The pressure force on face CD. =
(c). The shear force, τ × 2πr∆x on the surface of the fluid element. As there is no acceleration hence:
Net force in the x direction = 0
ΣFx = 0 results in
As ∂P/∂x across a section is constant, thus the shear stress τ varies linearly with the radius r as shown in the Figure.
Shear stress and velocity distribution in laminar flow through a pipe
r + y = R
dr + dy = 0
dr = − dy
u = 0 at r = R
& u = uMax at r = 0
Discharge, Q = A × uavg
Ratio of maximum velocity to average velocity
Thus, the Average velocity for Laminar flow through a pipe is half of the maximum velocity of the fluid which occurs at the centre of the pipe.
Pressure variation in Laminar flow through a pipe over length L
On integrating the above equation on both sides:
Head loss in Laminar flow through a pipe over length L
Above equation is hagen poiseuille equation.
As we know,
Radial distance from the pipe axis at which the velocity equals the average velocity
For steady and uniform flow, there is no acceleration and hence the resultant force in the direction of flow is zero.
∂τ/∂y = ∂P/∂x
Velocity Distribution (u):
the value of shear stress is given by:
∂τ/∂y = ∂P/∂x
Boundary condition, at y = 0 u = 0
Thus, velocity varies parabolically as we move in y-direction as shown in Figure.
Velocity and shear stress profile for turbulent flow
Discharge (Q) between two parallel fixed plates:
The average velocity is obtained by dividing the discharge (Q) across the section by the area of the section t × 1.
Ratio of Maximum velocity to average velocity:
Pressure difference between two parallel fixed plates
There are two correction factors:
1. Momentum correction factor (β)
It is defined as the ratio of momentum per second based on actual velocity to the momentum per second based on average velocity across a section. It is denoted by β.
2. Kinetic energy correction factor (α):
It is defined as the ratio of kinetic energy of flow per second based on actual velocity to the kinetic energy of the flow per second based on average velocity across a same section.
Let p = momentum
P = momentum/sec.
For flow through pipes values of α and β:
Turbulent flow is a flow regime characterized by the following points as given below
Shear stress in turbulent flow
In case of turbulent flow there is huge order intermission fluid particles and due to this, various properties of the fluid are going to change with space and time.
Average velocity and fluctuating velocity in turbulent flow
Similar to the expression for viscous shear, J. Boussinesq expressed the turbulent shear mathematical form as
where τt = shear stress due to turbulence
η = eddy viscosity
u bar = average velocity at a distance y from the boundary. The ratio of η (eddy viscosity) and (mass density) is known as kinematic eddy viscosity and is denoted by ϵ (epsilon). Mathematically it is written as
ε = η/ρ
If the shear stress due to viscous flow is also considered, then …. shear stress becomes as
The value of η = 0 for laminar flow.
Reynolds Expression for Turbulent Shear Stress.
Reynolds developed an expression for turbulent shear stress between two layers of a fluid at a small distance apart, which is given as:
τ = ρu'v'
where u’, v’ = fluctuating component of velocity in the direction of x and y due to turbulence.
As u’ and v’ are varying and hence τ will also vary.
Hence to find the shear stress, the time average on both sides of the equation
The turbulent shear stress given by the above equation is known as Reynold stress.
Prandtl Mixing length theory:
According to Prandtl, the mixing length l, is that distance between two layers in the transverse direction such that the lumps of fluid particles from one layer could reach the other layer and the particles are mixed in the other layer in such a way that the momentum of the particles in the direction of x is same.
In the above equation, the difference between the maximum velocity umax, and local velocity u at any point i.e. (umax - u) is known as ‘velocity defect’.
Velocity distribution in turbulent flow through a pipe
Laminar sublayer thickness
Laminar sublayer thickness