A thin layer adjoining the solid boundary, called the boundary layer, where the viscous effects and rotation cannot be neglected. An outer region away from the surface of the object where the viscous effects are very small and can be neglected. The flow behavior is similar to the upstream flow. In this case a potential flow can be assumed.

## Mathematical Models in Boundary Layer Theory - CRC Press Book

This velocity gradient in a real fluid sets up shear forces near the boundary that reduce the flow speed to that of the boundary. It is called the laminar sublayer. The vertical scale has been greatly exaggerated and horizontal scale has been shortened. The velocity profile in this layer is very nearly linear, and the flow is nearly parallel. Since there is no viscosity for the case of ideal fluid Fig. Figure 6. Then the reduction of mass flow rate is obtained as where b is the plate width.

The plate retards the oncoming stream greatly, and small changes in flow parameters cause large changes in the pressure distribution along the plate. Such thick-shear-layer flows are typically studied by experiment or by numerical modeling of the flow field on a computer 16 Comparison of flow past a sharp flat plate at low and high Reynolds numbers: a laminar, low-Re flow; b high-Re flow.

The turbulent flow formula applies for Rex greater than approximately In all cases these boundary layers are so thin that their displacement effect on the outer inviscid layer is negligible. At what distance x from the leading edge will the boundary layer thickness be 1 in?

If contradictory, try turbulent flow. It is only 10 percent higher than the known accepted solution for laminar flat-plate flow eq. When both sides of a thin plate are subjected to flow, A becomes the total area of the top and bottom surfaces. Compute the boundary layer thickness at the trailing edge for a air and b water at 68F.

The predicted laminar thickness is 29 The boundary layer thickness is 30 Example 2 Water in a city is distributed by extensive piping networks. Oil and natural gas are transported hundreds of miles by large pipelines. Blood is carried throughout our bodies by arteries and veins.

### Hurricane boundary‐layer theory

The cooling water in an engine is transported by hoses to the pipes in the radiator where it is cooled as it flows. This is because pipes with a circular cross section can withstand large pressure differences between the inside and the outside without undergoing significant distortion. Laminar flow is encountered when highly viscous fluids such as oils flow in small pipes or narrow passages.

This ratio is called the Reynolds number and is expressed for internal flow in a circular pipe as 38 Reynolds Number The hydraulic diameter is defined such that it reduces to ordinary diameter D for circular pipes, 40 Reynolds Number Because of the no-slip condition, the fluid particles in the layer in contact with the surface of the pipe come to a complete stop. The variation of wall shear stress in the flow direction for flow in a pipe from the entrance region into the fully developed region.

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This simplistic approach gives reasonable results for long pipes but sometimes poor results for short ones since it under predicts the wall shear stress and thus the friction factor. There is no motion in the radial direction, and thus the velocity component in the direction normal to flow is everywhere zero. There is no acceleration since the flow is steady and fully developed. Therefore, the drop of pressure from P1 to P2 in this case is due entirely to viscous effects.

The head loss is caused by viscosity, and it is directly related to the wall shear stress. Gravity helps downhill flow but opposes uphill flow. Therefore, much greater pressure differences need to be applied to maintain a specified flow rate in uphill flow although this becomes important only for liquids, because the density of gases is generally low.

The pressure at the pipe inlet and outlet are measured to be and 97 kPa, respectively. Also verify that the flow through the pipe is laminar.

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Determine a the head loss, b the pressure drop, and c the pumping power requirement to overcome this pressure drop. These fluctuations provide an additional mechanism for momentum and energy transfer. The very thin layer next to the wall where viscous effects are dominant is the viscous or laminar or linear or wall sublayer. The value of n increases with increasing Reynolds number. Although it is developed for circular pipes, it can also be used for noncircular pipes by replacing the diameter by the hydraulic diameter.

Determining the pressure drop or head loss when the pipe length and diameter are given for a specified flow rate or velocity 2. Determining the flow rate when the pipe length and diameter are given for a specified pressure drop or head loss 3.

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Determining the pipe diameter when the pipe length and flow rate are given for a specified pressure drop or head loss 88 Turbulent Flow in Pipes Determine the pressure drop, the head loss, and the required pumping power input for flow over a ft-long section of the pipe. Therefore, the flow is turbulent. Axisymmetric and Three—Dimensional Boundary Layers.

Unsteady Boundary Layers. Extensions to the Prandtl Boundary—Layer Theory. Onset of Turbulence Stability Theory. Fundamentals of Turbulent Flows. Internal Flows. Unsteady Turbulent Boundary Layers. Turbulent Free Shear Flows. Numerical Integration of the Boundary—Layer Equations. Back Matter Pages About this book Introduction This new edition of the near-legendary textbook by Schlichting and revised by Gersten presents a comprehensive overview of boundary-layer theory and its application to all areas of fluid mechanics, with particular emphasis on the flow past bodies e.

Ruhr University Bochum Insti. Bochum Germany.