Ghosh - 550Page 110/30/2018
Viscous Flows
This chapter discusses all real fluid flow applications in the realm of incompressible flows. Due to the limited time, compressible flow applications are covered in other courses such as Ideal Flows or Aerodynamics. Incompressible viscous flow applications may range from low to high speeds and internal and externalflow configurations. Internal flows occur when the fluid medium is surrounded by solid walls, which may be stationary or moving. On the other hand, for solids surrounded by fluid particles, we get the external flow configurations. A critical parameter in the flow study will be the non-dimensional quantity Reynolds number, Re. This number is the ratio of inertia and viscous forces in fluid flow. It is expressed as:
In the above expression, and are absolute and kinematic viscosities, U = Characteristic velocity, and L = Characteristic length. For example, for a steady external flow over a circular cylinder, U = Free stream velocity and L = Diameter of the cylinder. However, for steady internal pipe flow configurations,U= = Average velocity of the flow through a cross section.
We shall discuss flow based upon Re rather than speed. The advantage in doing that is high or low speed flows can be discussed at the same time in the context of low viscosity or highly viscous behavior. Another measure of flow speed can be by the amount of turbulence. For very high-speed flows, Re is high enough to call the flow fully turbulent. On the other hand, very low speed flows will be categorized invariably as laminar.
Implications of Viscosity
(Parallel vs. Fully-Developed Flows)
We earlier saw that viscosity manifests itself through creation of both shear and rotation. One of the most important characteristics of a real fluid is it satisfies “no slip” condition on a solid surface. In other words, whenever fluid flows past a solid surface, the layer of fluid in contact with the surface cannot slip against the surface, with the result that all components of fluid velocity will be zero at the solid,impermeable surface. This condition creates the highest shear on the fluid by a solid non-porous wall. As fluid particles adjacent to the wall try to stop the next layer of fluid, the shear gradually loses its strength as we move away from the wall. This is the cause of the boundary layer formation on a solid surface. We shall investigate this in much more depth in the next chapter on external flows. The internal flows discussed in this section willbe beyond the entrance length, which means that boundary layer growth from each wall has already met at the center of the channel.
Beyond the entrance length, which is typically 138-140 D for laminar pipe flows and 25-40 D for turbulent pipe flows, we call the flow fully developed. That is:
Parallel Flows:Instead of assuming fully developed flow, if we assume a parallel flow, it means all fluid streamlines are parallel. In such a case of parallel flow along x,
Since all fluid media must satisfy the mass, momentum and energy equation, we find by the application of continuity equation for incompressible flows,
We therefore find that a parallel flow is indeed fully developed.
With the introduction of parallel flows, simplification of the governing equations becomes much simpler. We now develop the applications for some specific types of internal flows.
Plane Poiseuille Flow
This is the case of fully developed incompressible flow between two infinitely large parallel plates. We seek the velocity profile and shear flow field for such flows. As before, if the flow is assumed parallel in the x-direction, . Therefore, the continuity equation reduces to , which satisfies the fully developed condition. Let us investigate the y- and z-momentum equations for such a flow. Also, we assume that the body forces are negligible. Therefore:
from above, which means that pressure is a function of x only. Now we simplify the x-momentum equation:
(Note that was modified to from the y and z equation results)
Let us further assume that the flow is steady. . Also . Furthermore, the flow can be assumed to be free from the end conditions since the plates are infinitely long and deep., which means also. Thus the x-equation simplifies to:
We can integrate this equation twice in y to write:
(C1 and C2 = Constants)
Boundary Conditions:Since both plates are stationary, u(0) = 0, u(h) = 0
The velocity profile u(y) may be evaluated with, and,
To be able to plot this velocity profile, let us assume
Note that the velocity profile starts with a zero value on the wall, reaches a peak value of 6.25 m/s in the middle of the channel (h = 0.5 m) before reducing to zero on the upper wall (h = 1 m) symmetrically. Also, try to plot the function when (instead of –5 N/m3). You will see an unrealistic curve (showing fluid bulges out along "-" x direction). We can check the volumetric flow rate to claim this point.
, where w = depth of the channel and
or,
From this expression, it is easy to see that since Q, w, h, and are all positive quantities, Q cannot be positive unless . Thus, we make an important discovery for Plane Poiseuille Flow: A Plane-Poiseuille flow cannot exist if the pressure gradient, , is not negative. We also introduce a new definition of average velocity in this context. Average velocity through any area A is defined as the volumetric flow rate per unit depth,i.e.
If we evaluate the maximum velocity in this flow,
, which occurs at the center of the channel.
Therefore we notice that the maximum velocity
or, for this flow.
Shear Stress Distribution:
If we plot this function along with the velocity profile, we notice a linear variation of shear stress and shear force as follows:
These plots were made with as stated before.
Couette Flow
This type of flow is also between infinite parallel plates. However, the boundary conditions are a little different from Plane Poiseuille Flows. Here one of the plates remains stationary, whereas the other moves with a constant velocity, U. For visualization, we assume the bottom plate stationary and the top plate moving.
All the assumptions applicable to the derivation of Plane Poiseuille flows hold in the case of Couette flows. Thus, we may skip part of the derivation and start with the velocity profile.
Now,
If we compare the above velocity profile with that obtained for Plane Poiseuille flows, we find the right hand side has an additional term, . The plot of just this term is a linear velocity profile from y = 0, u = 0 to y = h, u = U. Thus, the Couette flow velocity profile may be thought of as the superposition of the Plane Poiseuille flow’s velocity profile and this additional linear profile. Because of this additional fluid momentum, Couette flows can exist even with mild adverse pressure gradient (i.e., ). Recall that the existence of Q > 0 makes the flow possible.
Since we know the velocity profile , all the flow quantities such as volumetric flow rate, average velocity, maximum velocity, shear stress and shear force distributions can be computed as before using their respective formulae.
Hagen Poiseuille Flow (or, Pipe Flow)
Now we come to derive the most popular application of the internal flows, commonly known as Hagen Poiseuille Flow or, simply pipe flows. Since pipes have cylindrical geometry, we use the cylindrical form of the momentum equations. Let us assume an incompressible, steady flow through a circular pipe without any appreciable body forces. Assuming a parallel flow in the z-direction, , but .
Continuity equation
As in the case of Plane Poiseuille flow, writing out the momentum equations in and r direction will simply result in . Therefore, let us focus on z-direction.
We can further assume because of the cylindrical symmetry.
[]
or, integrating twice over “r”, we get
(C1, C2 = Constants)
Since the pipe radius is R, the boundary conditions may be written as
and .
The second boundary condition is due to flow symmetry at r = 0, whereas the first one is due to “no-slip” condition. Solving the constants C1and C2we get
As in the case of Plane Poiseuille flows, for this flow to exist (i.e., Q > 0).
Some additional results are:
, , , and
[Note: You must use an annular area element to derive V and Q results.]
Velocity Profiles in Pipes
The laminar velocity profile that we just derived is parabolic in shape, given by
(Vzmax = Constant)
However, as the flow Reynolds numbers increase beyond Recr = 2300, the fluid flow in pipes can be considered turbulent. Turbulent velocity profiles are much flatter compared to the laminar profiles because of more friction near the walls. These are given by a “power law”:
where, 6 < n < 10. The larger the Reynolds number, the higher the value of n. The most popularly used velocity profile for turbulent fully developed flows is for n = 7.
Also due to the flatter profiles
the turbulent flow is closer to 1.0. The power law profiles satisfy
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