Consider viscous flow over a flat plate a. Write the definition of the Rex , based on distance, x, from the leading edge of a flat plate, and explain its significance b. Sketch the development of the velocity boundary layer height, d, beginning at the plate’s leading edge, and extending into the turbulent region, and indicate the range of the Reynolds number in the relevant regions c. How does d change with distance x in the laminar region.

Explanation:

Solution:-

- To categorize the flow conditions of any fluid we utilize a dimensionless number, called Reynold's number ( Re ) to study the behaviour of the fluid.

- Reynold's number is proportional to the ratio of inertial forces ( forces that resist any change in motion of a unit mass ) and viscous forces ( forces that resist any iner-plane deformations between layers of fluid ).

- Considering 2-Dimensional viscous flow over a flat plate, the Reynold number (Re) is mathematically expressed as a function of distance "x" denoted from leading edge and along the length of the plate:

Where,

               U: The free stream velocity of the fluid

               ν: The kinematic viscosity of the fluid

- The distance "x" along the length of the plate is substituted in the above formula and the corresponding Reynold number is evaluated. This gives a highly localized value about "x".

- The purpose of the Reynold number is the substitution of dynamically similar fluids i.e Fluid with the same Reynold's number when testing models to see how they would behave in a specific environment.

- The Reynolds number has a set of ranges above and below the critical range defined by the critical length "xc" along the plate.  The range below the critical has laminar flow characteristics, whereas the range above the critical has turbulent flow. The laminar region has flow along smooth streamlines, while turbulent region is characterized by 3 - dimensional random eddy.

- The critical length " xc " is determined from the critical Reynold number i.e ( 5 x 10^5 ) which is a small region that has mixed characteristics of laminar and turbulent conditions.

- The flow at the boundaries has zero velocity, there is a steep velocity gradient from the boundary into the flow. This velocity gradient in a real fluid sets up shear forces near the boundary that reduce the flow speed to that of the boundary. That fluid layer which has had its velocity affected by the boundary shear is called the "boundary layer"

-  For smooth upstream boundaries, " x << xc " or " Re_x < 5 x 10^5 " , the boundary layer starts out as a laminar boundary layer in which the fluid particles move in smooth layer.

- As the laminar boundary layer increases in thickness, it becomes unstable as changes in motion become more predominant ( inertial ) than viscous effect of fluid layers. This leads to a transformation of laminar boundary layer into turbulent boundary layer in which fluid particles move in haphazard paths.  " x > xc " or " Re_x > 5 x 10^5 "

- The boundary layer thickness/height d increases as x increases. The relationships for laminar and turbulent regions of boundary layer are given as follows:

- To construct a function of boundary layer thickness " d " and length from leading edge of the plate " x ". We use the Re_x relation and substitute, we get the following proportionalities for our sketch:

                  d ∝ √x  .... Laminar region

                  d ∝  .... Turbulent region

- Use the above relation to develop sketch for the boundary layer along the length "x" from leading edge.

- The sketch is given as an attachment.

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