Fluid Mechanics Fundamentals | Streamlines, Flow – PDF

Fluid mechanics Fundamentals of flow 1 Two methods describes the movement of flow, which are 1. The Lagrangian method. Arbitrary particle follows its kaleidoscopic changes in velocity and acceleration. 2. The Eulerian method. Changes in velocity and pressure are studied at fixed po...

Fluid mechanics Fundamentals of flow 1 Two methods describes the movement of flow, which are 1. The Lagrangian method. Arbitrary particle follows its kaleidoscopic changes in velocity and acceleration. 2. The Eulerian method. Changes in velocity and pressure are studied at fixed positions in space x, y, z and at time t. 2 Streamline and stream tube Streamline is a curve formed by the velocity vectors of each fluid particle at a certain time. The curve where the tangent at each point indicates the direction of fluid at that point is a streamline. Floating powder on the surface of flowing water gives the flow trace of the powder as shown in the following figure (a). A streamline is obtained by drawing a curve following this 3 flow trace. From the definition of a streamline, since the velocity vector has no normal component, there is no flow which crosses the streamline. Considering two-dimensional flow where the gradient of the streamline is dy/dx, then the following equation of the streamline is obtained dx/u = dy/υ (1) where u is the velocity in the x-direction. υ is the velocity in the y-direction. When streamlines around a body are observed, then the streamlines vary according to the relative relationship between 4 the observer and the body. The lines which show streams include the streak line and the path line (figure (b)).  The streak line is the line formed by a series of fluid particles which pass a certain point in the stream one after another. The path line is observed in figure (c). On the other hand, by the path line is meant the path of one particular particle starting from one particular point in the stream. In the case of steady flow, the above three kinds of lines all happen together. 5 6 By taking a given closed curve in a flow and drawing the streamlines passing all points on the curve, a tube can be formulated.  This tube is called a stream tube. Since no fluid comes in or goes out through the stream tube wall, the fluid is regarded as being similar to a fluid flowing in a solid tube.  This assumption is convenient for studying a fluid in steady motion. 7  Steady flow and unsteady flow If a flow state is expressed by velocity, pressure, or density at any position, and does not change with time, then this flow is called a steady flow. As an example, if water runs out of a tap while the handle is being turned then the flow is an unsteady flow. If a flow state does change with time, then is flow is known as an unsteady flow. For example, when water runs out a tap while the handle is stationary, leaving the opening constant, then the flow is steady. 8  Three-dimensional, two-dimensional, and one-dimensional flow All general flows have velocity components in x, y and z directions.  The flow is called three-dimensional flow. Expressing the velocity components in the x, y and z axial directions as u, ʋ and w, then u = u (x, y, z, t ) (2-a) υ = υ (x, y, z, t ) (2-b) w = w (x, y, z, t ) (2-c) 9 Consider water running between two parallel plates.  Cross-cut vertically to the plates.  Parallel to the flow.  If the flow state is the same on all planes parallel to the cut plane, then the flow is called a two dimensional flow since it can be described by two coordinates x and y. The velocity components in the x and y directions are expressed as u and ʋ, then u = u (x, y, t) (3-a) υ = υ (x, y, t) (3-b) 10 Consider water flowing in a tube in terms of average velocity, then the flow has a velocity component in the x direction only. One-dimensional flow is a flow whose state is determined by one coordinate (x only). One-dimensional flow velocity (u) depends on coordinates (x) and (t) only u = u (x, t) (4) 11  Laminar and turbulent flow  Osborne Reynolds studied the states of flow.  Reynolds used the set up to the right.  Where coloured liquid was led to the entrance of a glass tube.  As the valve was gradually opened, the Coloured liquid flows without mixing with peripheral water, as shown in figure (a).  This is known as the laminar flow. 12 When the flow velocity of water in the tube reached a certain value, then the line of coloured liquid become turbulent on mingling with the peripheral water (figure b).  This is known as the turbulent flow. The flow velocity at the time when the laminar flow had turned to turbulent flow is called the critical velocity. 13  Reynolds number Reynolds discovered that a laminar flow turns to a turbulent flow when the value of the non-dimensional quantity ρʋd/μ reaches a certain amount whatever the values of the average velocity (ʋ), the glass tube diameter (d), water density (ρ) and water viscosity (μ ). Thus Reynolds number (Re) is Re = ρ υ d / μ = υ d / ν (5) where ν is the kinematic viscosity. 14 Whenever the velocity is the critical velocity (ʋc), then the critical Reynolds number is obtained Rec = υc d / ν (6) The value of Reynolds number is affected by the turbulence existing in the fluid coming into the tube. Reynolds number at which the flow remains laminar, is called the lower critical Reynolds number.  This value is said to be 2320. 15  Incompressible and compressible flow  Liquid is an incompressible fluid.  Gas is a compressible fluid.  In the case of a liquid, compressibility is taken into account whenever the liquid is highly pressurised, such as oil in a hydraulic machine.  In the case of a gas, the compressibility may be ignored whenever the change in pressure is small.  As a standard for this judgement, Δρ/ρ or the Mach number (M) is 16 used.  Mach number When a fluid flows at high velocity, or when a solid moves at high velocity in a fluid at rest, the compressibility of the fluid can dominate so that the ratio of the inertia force to the elasticity force is then important. (7) where a is the velocity of sound. K is the bulk modulus. L is the length. ρ is the velocity potential. 17 υ is the velocity (y-direction). In this case, the square root is selected to be defined as the Mach number (M) M=υ/a (8) M < 1 then the flow is called subsonic flow. M = 1 then the flow is called sonic flow. M > 1 then the flow is called supersonic flow. When M = 1 and M < 1 and M > 1 zones are simultaneous (parallel), then the flow is called transonic flow. 18  Rotation and spinning of a liquid Fluid particles running through a narrow channel flow, while undergoing deformation and rotation. Assume that (figure to the right) an elementary rectangle of fluid ABCD with sides dx and dy.  Which is located at O.  At time (t) moves to O’.  While deforming to A’B‘C’D’ (time dt). 19 AB in the x direction moves to A‘B’ while rotating by dɛ1. AD in the y direction rotates by dɛ2. Thus dɛ1 = (∂υ/∂x) dx dt (9) dθ1 = dɛ1/dx = (∂υ/∂x) dt (10) dɛ2 = - (∂u/∂y) dy dt (11) dθ2 = dɛ2/dy = - (∂u/∂y) dt (12) The angular velocities of AB and AD are ω1 and ω2 20 ω1 = dθ1/dt = ∂υ/∂x (13) For centre (O), the average angular velocity (ω) is ω = (1/2) (ω1 + ω2) = (1/2) (∂υ/∂x – ∂u/∂y) (15) The above equation can be written as follows ζ = ∂υ/∂x – ∂u/∂y (16) Equation 16 gives what is called the vorticity for the z axis. ∂υ/∂x – ∂u/∂y = 0 (17) Ir-rotational flow is the case where the vorticity is zero. 21 As shown in the figure to the right, a cylindrical vessel containing liquid spins about the vertical axis at a certain angular velocity. The liquid makes a rotary movement along the flow line. At the same time, the element itself rotates. 22 Figure to the right shows the case of rotating flow which is observed whenever liquid is made to flow through a small hole in the bottom of a vessel. The liquid makes a rotary movement, its microelements always face the same direction without performing rotation. This case is a kind of ir-rotational flow is called free vortex flow. 23  Circulation Assuming a closed curve (s), the integrated velocity (ύs) along this same curve is called the circulation (Γ). The integrated velocity (ύs) is the velocity component in the tangential direction of the velocity (υs) at a given point on this curve. 24 Counter-clockwise rotation is taken to be positive, with the angle between υs and ύs as θ, then Γ = ʃ ύs ds = ʃ υs cosθ ds (18) Divide the area surrounded by the closed curve (s) into micro- areas by lines parallel to the x and y axes, and study the circulation dΓ of one such elementary rectangle ABCD (area dA) the following is obtained (19) 25  Vorticity (ζ) is two times the angular velocity (ω) of a rotational flow (equation 15).  The circulation is equal to the product of vorticity by area.  Integrating equation 19 for the total area, where the integration on each side cancels, leaving only the integration on the closed curve (s), thus the following is obtained Γ = ʃ ύs ds = ʃ ζ dA (20)  From equation (20) it is found that the surface integral of vorticity (ζ) is equal to the circulation.  26 This is known as Stokes’ theorem. If there is no vorticity inside a closed curve → then the circulation around it is zero. Stokes’ theorem is utilised in fluid dynamics to study 1. The flow inside the impeller of pumps and blowers. 2. The flow around an aircraft wing. 27