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boundary layer

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1
Contents
1 Abstract ............................................................................................
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2 General Symbols ............................................................................................................................. 1 3 Introduction .................................................................................................................................... 2 3.1 Aims and Objective ................................................................................................................. 2 4 Literature Review ............................................................................................................................ 3 4.1 History of Fluid Dynamics ....................................................................................................... 3 4.2 Introduction to Boundary Layer .............................................................................................. 4 4.3 Background Theory ................................................................................................................. 5 4.3.1 Basic physical concepts ................................................................................................... 5 4.3.2 Theoretical Foundation ................................................................................................... 8 4.3.3 Effects of Boundary Layer ............................................................................................. 21 4.3.4 Solutions of Boundary Layer Problem........................................................................... 24 4.4 Flow Separation and Flow control ...........................................
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4.5 CFD ........................................................................................................................................ 36 4.6 Ansys ........................................................................................
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5 Simulation ..................................................................................................................................... 37 6 Results ........................................................................................................................................... 37 7 Conclusions ................................................................................................................................... 37 8 Acknowledgement ........................................................................................................................ 37 9 Bibliography .................................................................................................................................. 37 10 Appendix ................................................................................................................................... 38
1
General Symbols
Density of fluid
Reynolds number
Viscosity
Kinematic viscosity u Velocity in x-direction
2 v Velocity in y-direction w Velocity in z-direction
Cofficient of drag D Drag force
∗
Displacement Thickness
Momentum Thickness
Shear Stress
2
Introduction
All the man-made vehicles have to travel through a viscous fluid when it moves from one point to another. This is true for no matter what the vehicle was designed for, i.e. flying, moving in the sea or moving on a road. The fluid flow behaviour close to the surface of a vehicle is of great importance when designing the outer body of a vehicle as it determines the amount of resistance the vehicle has to overcome when moving. Fluid dynamics is the study of fluid in motion which started a long time ago as the humans started to understand the effect of the viscous flow on the projectile weapons and pipes used to transport fluids. This report concentrates on the aerodynamics aspect of the fluid dynamics and the first part of this report is about an important breakthrough in the study of fluid dynamics made by Ludwig Prandtl in 1904 through his discovery of thin boundary layer
and it’s
implication. He discovered that for a moving fluid with low viscosity a boundary layer is formed which confines the viscous effect. The discovery of boundary layer allowed some simplification of the Navier-Stokes equation that governs the fluid flow which existed and was derived before he discovered the formation of boundary layers. The first part looks at some of the solution of the laminar boundary layer prolem. Second part of this report studies look at the turbulent flow regime and the effect of the turbulent flow. Some flow control method in order to avoid turbulence is also looked at. Finally, flow over a flat plate CFD simulation was done by using software called Ansys has been carried out in order visualise boundary layer formation over a flat plate.
2.1
Aims and Objective
The main aims and objective and aim of this report are as follows:-
Look at the theories associated with boundary layer.
Study the effect of boundary layer.
Explore some of the methods employed by engineers to simplify or solve laminar boundary layer problems.
Study the effect of active flow control techniques.
3
Use of CFD software as Ansys to simulate boundary layer problem.
3
Literature Review
3.1
History of Fluid Dynamics
The study of viscous fluid flow started at prehistoric times as human began developing weapons that were streamlined, such as weighted spears and fin-stabilised arrow. This implies that the primitive people recognised the effect of viscous resistance and took the first step in solving the viscous flow problem. (White) Later, Greek mathematician Archimedes (287-212 B.C) gave the exact solution for the problem of viscous fluid at rest. At the same time the Romans displayed intuitive understating of the effect of viscous resistance through their infrastructure and water-supply systems. (White) After a period of little or no progress in the study of fluid flow, Leonardo da Vinci deduced the equation of conservation of mass for incompressible one-dimensional viscous flow in 1500. (White) Another notable achievement was made by Evangelista Torricelli by publishing his theorem in 1964 which states that
“the velocity of efflux of a (viscous) liquid from a hole in a tank is equal to the velocity which a liquid particle would attain in free fall”.
(White) The achievements mentioned so far were a result of study of inviscid flow or the flow of perfect fluid which happened to be valid for invisicd or real flow situation too and can be considered indirect study of viscous flow. The first direct study was done by Edme Mariotte who invented a balance system to measure the drag of a model held inside a wind tunnel. (White)
In 1687 Newton published his “Principia” which described
the viscous behaviour of the majority of the fluids. (White) In 1738, Daniel Bernoulli Demonstrated the proportionality between pressure gradient and the acceleration in the in viscid flow. Euler then derived the frictionless equation which is now knows as the
Bernoulli’s
equation. This derivation is still unchanged in ideal-fluid theory. At the same time
period Jean d’Alembert published his famous paradox in 1752, which states that “a body immersed in a frictionless flow would have zero drag”.
(White) Theoretical result of the Paradox caused the fluid mechanics to divide into two parts: hydrodynamics which was the invsicid flow study consisting mathematical formulas and the hydraulics which relied on experimental measurements. (White) After the Derivation of frictionless equation, the frictional resistance term was added to the equation through the contribution of 5 people. Those 5 people were Navier. Cauchy, Poisson, St. Venant and Stokes. In the present time, these equations are called the Navier-Stokes equation and are fundamental to the fluid dynamic analysis. This Navier-Stoke equation is non-linear, complex and difficult to solve with little particular solution discovered so far. (White) The biggest breakthrough for practical engineering was done by Ludwig Prandl in 1904 through his discovery of boundary layer which confined the vicious effect. This discovery allowed simplification
4 of the formidable Navier-Stoke equation in some circumstances through use of boundary conditions. It also allowed engineers to look at the fluid flow problem intelligibly. (White)
3.2
Introduction to Boundary Layer
Due to the theoretical result of the famous paradox by Jean d’Alembert,
the study of fluids branched off into two seemingly different field of study. On one hand, there was the study of hydrodynamics which resulted in the development of the inviscid theory consisting of mathematical equation. The invisicid theory produced relevant results when the focus was not on the flow regions close to a solid body. However, same could not be said about the region of the flow near a solid body as the theory ignored the frictional resistance and the drag associated with it. On another hand, there was the study of hydraulics which focused on the mechanical properties of liquids and was based on formulae and datas generated through experiments and experience of engineers. This field of study or approach focused on the engineering use of the fluid properties and helped solve engineering problems such as design. (YOUNG) Even though, these two fields seemed remarkably different, Prandtl linked these two fields by using his boundary layer theory. The theory was based on certain basic observations. Those observations were as follows:- (YOUNG)
No matter how small the viscosity of a moving fluid is, it cannot be ignored. The fluid at the surface of a body is at rest and the relative fluid velocity increases with distance normal to the surface until it equals to the free stream velocity . (YOUNG)
The viscosity of a fluid results in shear stress which is directly related to the velocity gradient due the difference in the relative velocity of the fluid or the rate of strain. (YOUNG)
The Reynolds number is a controlling parameter of flow phenomena and is the ratio of inertial and viscous forces.
(YOUNG)
The relative thickness of boundary layer decreases with increase in Reynolds number. So, high Reynolds number permits some simplification to the Navier-Stokes equation or the equation of motion of a viscous fluid. (YOUNG) From these observations it can be deduced that when there is an interaction between a viscous fluid and a solid body in motion, a region where relative velocity of the fluid increase from zero at the surface of the body to free stream velocity at the boundary of the boundary layer is created. In other words a region where the effect of the viscous force can be felt because of the presence of velocity gradient is created. So, a boundary layer can be described as a boundary confining the viscous effect or the region where the viscous effect cannot be ignored. The last observation implies that the boundary layer thickness is related to the Reynolds number or the fluid velocity. The thinness of the boundary layer at higher Reynolds number can be explained by the fact that higher speed allows lower time for the diffusion of viscous effect. A visual representation of the boundary layer formation is shown in Figure 1.

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