In this work, a model for the hydrodynamic drag force on a steadily translating circular cylinder was studied for Reynolds number, Re << 1. The Navier-Stokes and continuity equations were solved to obtain the drag force and drag coefficient. Furthermore, the drag coefficient was plotted against varied values of Reynolds number, results and conclusions were made between the relationship.
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND TO THE STUDY.
A major objective of fluid mechanics is to understand how impermeable and rigid surfaces such as pipes and particles affect the flow of a fluid and the effect of the fluid on the particles. One of the effects of fluid on particles in which they flow is the drag exerted by the fluid on the particles.
A body moving through a fluid, experiences a drag force which is usually divided into two components: frictional (viscous) drag and pressure (form) drag. Frictional drag results from the friction between the fluid and the surfaces over which it is flowing. This friction is associated with the development of boundary layers. The pressure drag comes from the eddying motions that are set up in the fluid by the passage of the body. This pressure or form drag is associated with the formation of a wake and it is usually less sensitive to Reynolds number than the frictional drag. The two types of drag are due to viscosity (if the body was moving through an inviscid fluid, there would be no drag at all) but the distinction is necessary because the two types of drag are due to different flow phenomena. Frictional drag is important for attached flows (there is no separation) and it is connected to the surface area exposed to the flow. Pressure drag is important for separated flows and it is connected to the cross-sectional area of the body.
There are also two kinds of bodies: streamlined bodies and blunt (bluff) bodies. The two kinds are differentiated by the drag that is dominant in their flow. When the drag is dominated by viscous drag, we say the body is streamlined and when it is dominated by pressure drag, we say the body is blunt. The drag that dominates the flow depends entirely on the shape of the body. A streamlined body looks like a fish or an aerofoil at small angles of attack and the streamlines of a streamlined body are smoothly around the body. A blunt body looks like a brick, a cylinder, a sphere or an aerofoil at large angles of attack. The streamlines of a blunt body break away whenever a sharp change in direction occurs. For streamlined bodies, frictional drag is the dominant source of resistance while for a blunt body, pressure drag is dominant. For a given frontal area and velocity, a streamlined body will always have a lower resistance than a blunt body. Cylinders and spheres are considered blunt bodies because at large Reynolds numbers, the drag is dominated by the pressure losses in the wake. The major difference between these two kinds of bodies is that in streamlined flow, the regions where losses occur are inside the boundary layers and wakes remain reasonably thin, whereas in a blunt (bluff) body, adverse pressure gradients cause the boundary layers to separate which creates a large wake filled with energetic eddies which increases the drag on the body.
Drag is an enemy of aircraft, submarines, etc. Its calculation helps in the design of structures like automobiles, towers, etc. Its reduction helps to increase the speed of the objects and lowers fuel consumption as they move in the fluid. Drag coefficient is a dimensionless quantity used to quantify the drag or resistance of an object in a fluid environment, such as air or water. For instance, a lower drag coefficient indicates the object will have less resistance. For instance, the design of cars has evolved from 1920s to the end of the 20th century. This change in design from a blunt body to a more streamlined body reduced the drag coefficient from about 0.95 to 0.30 (en.m.wikipedia.org/wiki/Oseen equations, March 2018). Drag coefficient is a function of speed, flow direction, object position, object shape and size, fluid density and fluid viscosity.
In this work, we model an equation for the drag force on a circular cylinder as it translates in a fluid.
A lot of work has been done on this subject by method of rigorous asymptotic analysis. In this work, we attempt to remove the mathematical complexity of asymptotic analysis by solving the Navier – Stokes equations directly to obtain the fluid velocity and then proceed to obtain the drag force and finally the drag coefficient which is a function of the Reynolds number. The drag coefficient is graphed against Reynolds number, results and conclusions are drawn from the relation.
1.2 STATEMENT OF THE PROBLEM
In fluid dynamics, drag, a type of friction, sometimes called fluid resistance is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. This can exist in a fluid/fluid presentation or in a fluid/solid interface. The importance of drag can therefore not be over emphasised especially in designs. The accurate assessment of drag results in economic design of automobiles, chimneys, towers, buildings, hydraulic structures etc. and therefore requires a clear understanding and an easy way to calculate it without mathematical complexities.
1.3 AIM AND OBJECTIVES OF STUDY
The aim of this work is to design a mathematical model for the drag force on a steadily translating circular cylinder in a fluid. The specific objectives are:
(i) Solve the Navier-Stokes equations to obtain the fluid velocity.
(ii) Find the drag force.
(iii) Find the drag coefficient.
1.4 SIGNIFICANCE OF STUDY
The demand by travellers all over the world for high speed, low energy consuming, time saving and efficient automobiles, train, aircrafts, ships etc., has gone up over the years. Also the occurrence of high speed and strong winds like the hurricanes, tornadoes, cyclones etc., has also ignited the desire for durable structures like towers by real estate developers. It then become imperative for designers, manufacturers, builders to have knowledge of drag and an easy way of calculating it. This knowledge is also a powerful tool in the hands of laboratory scientists, who need to compute the drag force on a solute by a solution of viscosity for the calculation of the rate of diffusion of the molecules of the solute.
1.5 SCOPE OF STUDY
This research work is restricted to the two dimensional Navier-Stokes equation as contained in literature.
1.6 DEFINITION OF TERMS
Fluid: All materials deform under the action of forces. An elastic material regains its original form when the applied force is removed, a plastic material leaves a permanent deformation with the removal of the applied force. A fluid is a material that deforms continuously without limit under the action of forces, however small they might be. It falls into two categories, namely liquids and gases. Fluid mechanics is the study of the motion of fluids.
Reynolds Number ( Re ): This is the ratio of the inertial forces to the viscous forces in a fluid. It is a dimensionless quantity that determines the characteristics of flow of a flow field, for instance whether a flow will be laminar or turbulent. Generally, Re is defined as
where U and L are velocity and length scales respectively.
is called the kinematic viscosity of the fluid. For a pipe flow,
Where d is the diameter of the pipe.
Steady Flow: This is a flow in which the fluid velocity is independent of time (t). The derivative of the fluid velocity with respect to time is zero. That is
Drag Force ( FD ): It is a frictional force that a fluid exerts on an object which acts in opposite direction to the motion of the object. It is a factor that determines the speed of an object in a fluid. For instance, the drag force on a sphere is
where a is the radius of the sphere moving with a speed of U sp
and m is the fluid viscosity.
Drag Coefficient ( CD ): This is the ratio
where r is the density of the fluid, U is the flow speed of the object relative to the fluid and A is the reference area of the object. Again for a sphere,
where the Reynolds number, 𝑅𝑒, is defined as:
It should be noted that the reference area depends on what type of drag coefficient is being measured. For automobiles and many other objects, the reference area is the projected frontal area of the vehicle and for aerofoil, the reference area is the wing area. Airships and some bodies of revolution use the volumetric drag coefficient, in which the reference area is the square of the cube root of the airship volume. Submerged streamlined bodies use the wetted surface area as the reference area. Two objects having the same reference area moving at the same speed through a fluid will experience a drag force proportional to their respective drag coefficients.
Incompressible Fluid: This is a fluid in which the density (r) is constant throughout its volume. That is
Examples of incompressible fluid are air and water.
Boundary Layer: In the flow of fluid around solid walls, it is observed that the fluid adheres to the wall of the solid showing that frictional forces (viscosity) retard the motion of the fluid in a thin layer near the wall. In that thin layer the velocity of the fluid increases from zero at the wall (no slip) to its full value at the external frictionless flow. This thin layer is called boundary layer.
Streamline: This is an imaginary curve drawn in a fluid so that its tangent at each point is the direction of the fluid velocity at that point. In Cartesian- coordinates, the equation
of a streamline is
u , v and w are the x, y and z components of the fluid velocity respectively.
Stream Function: This is a function of space and time, such that its partial derivative with respect to any direction gives the velocity component at right angles to this direction. For two dimensional steady flow, the stream function is defined as
In polar coordinates
Vorticity: Let q = ui + vj + wk be a fluid velocity such that curl q ¹ 0 .
The vector
is called vorticity vector. A flow is said to be irrotational when the vorticity vector of every fluid particle is zero, otherwise it is called a rotational flow.
Laminar Flow: A laminar flow is one in which the path lines of fluid particles do not intersect one another and move along a well -defined path. Examples of laminar flow are:
(i) Ground water flow.
(ii) Flow through a capillary.
(iii) Blood flow in veins and arteries.
Doublet: A source is an abstract point from which a fluid flows outward radially, symmetrical in all directions in the reference plane. Though a source does not occur in nature, the idea helps to describe many fluid motions as due to sources which are outside the boundaries of the fluid which we consider. A source is also a point at which fluid is continuously created and distributed. A sink is a negative source; it is a point of inward radial flow at which fluid is absorbed continuously. A doublet is a combination of an infinite source and sink at an infinitesimal distance.
Viscous: An infinitesimal fluid element is acted upon by two types of forces, namely, body forces and surface forces. The surface force on a fluid element is resolved into two components. The normal component to the fluid element is called normal stress or pressure while the tangential component is called the shear stress. A fluid is said to be viscous when normal and shearing stress exist. On the other hand, an inviscid fluid does not exert any shearing stress, whether at rest or in motion.
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