Lecture 4 Dr. Dhafer A .Hamzah Ch.5. Viscous flow: pipes and channel

Objectives concept of the Reynolds number developed; characteristics that distinguish laminar from turbulent flow are presented, and the categorization of flows into internal versus external is established; concentrates on internal-flow cases; steady, laminar, incompressible flows are first developed, since the losses can be computed analytically. .

Laminar and turbulent flows, internal and external flows The Reynolds number Laminar flow is defined as flow in which the fluid moves in layers,or laminas, one layer gliding smoothly over an adjacent layer with only a molecular interchange of momentum. Any tendencies toward instability and turbulence are damped out by viscous shear forces that resist relative motion of adjacent fluid layers . Turbulent flow has very erratic motion of fluid particles, with a violent transverse interchange of momentum.

The nature of the flow, i. e The nature of the flow, i.e., whether laminar or turbulent, and its relative position along a scale indicating the relative importance of turbulent to laminar tendencies are indicated by the Reynolds number. Two flow cases are said to be dynamically similar when: 1) They are geometrically similar, i.e., corresponding linear dimensions have a constant ratio. 2) The corresponding streamlines are geometrically similar, or pressures at corresponding points have a constant ratio.

In considering two geometrically similar flow situations, Reynolds deduced that they would be dynamically similar if the general differential equations describing their flow were identical. By changing the units of mass, length, and time in one set of equations and determining the condition that must be satisfied to make them identical to the original equations, Reynolds found that the dimensionless group ρul/μ must be the same for both cases. The quantity u is a characteristic velocity, l a characteristic length, ρ the mass density, and μ the viscosity. This group, or parameter, is now called the Reynolds number R, R= ρul/μ.

  Reynolds experiment:-to determine the significance of the dimensionless group, Reynolds conducted his experiments on flow of water through g1asstubes (Fig.5.1). A glass tube was mounted horizontally with one end in a tank and a valve on the opposite end. A smooth bellmouth entrance was attached to' the upstream end, with a dye jet so arranged that a fine stream of dye could be ejected at any point in front of the bellmouth. Reynolds took the average velocity Vas characteristic velocity and the diameter of tube D as characteristic length, so that R= ρuD/μ.factor).

For, small flows the dye stream moved as a straight line through the tube, showing that the flow was laminar. As the flow rate increased, the Reynolds number increased, since D, ρ, μ, were constant and V Was directly proportional to the rate of flow. With increasing discharge a condition was reached at which the dye stream wavered and then suddenly broke up and was diffused throughout the tube. The flow had changed to turbulent flow with its violent interchange of' momentum that had completely disrupted theorderly movement of laminar flow.

Starting with turbulent flow in the glass tube, Reynolds found that it always becomes laminar when the velocity is reduced to make R less than 2000.This is the Reynolds lower critical number for pipe flow and is of practical importance. With the usual piping installation, the flow will change from laminar to turbulent in the range of Reynolds numbers from 2000 to 4000. For the purpose of this treatment it is assumed that the change occurs at R =2000.

Internal and External flows Another method of categorizing flows is by examining the extent of the flow field. Internal flow involves. flow in a bounded region, as the name implies. External flow involves fluid in an unbounded region in which the focus of attention is on the flow pattern over a body immersed in the fluid.

At section A - A, near a well-rounded entrance, the velocity profile is almost uniform over the cross section. The action of the wall shearing stress is to slow down the fluid near the wall. As a consequence of continuity, the velocity must increase in the central region. Beyond a transitional length L' the velocity profile is fixed since the boundary influence has extended to the pipe centerline. The transition length is a function of the Reynolds number; for laminar flow Langhaar developed the theoretical formula:   𝐿 ′ 𝐷 =0.058 𝑅

Non dimensional groups which determine flow similarity Reynolds number Fig.8