REAL FLUIDS OUTLINE Introduction to Laminar and Turbulent Flow Reynolds Number Laminar Flow Flow in Circular Pipes Hagen Poiseuille Equation Flow between parallel plates
REAL FLUIDS Turbulent Flow About turbulent flow Head loss to friction in a pipe Shear stress in circular pipes Other head losses Total head and pressure line
Inviscid and Viscous Flows Inviscid Flows One in which viscous effects do not significantly influence the flow. Cases where the shear stresses in the flow are small and act over such small areas that they do not significantly affect the flow field. Examples are flows exterior to a body. Viscous Flows Viscous effects cause major losses; account for energy used in transporting fluids in pipelines. Caused by the no-slip condition (velocity is 0), and the resulting shear stresses. Examples: broad class of internal flows in pipes and in open channels.
External Flow Inviscid flow Boundary layer
Internal Flow u(x, y) Fully developed flow x L E (entrance length) u(y)
LAMINAR FLOW
Laminar versus Turbulent Flow v (t) v (t) t t v (t) v (t) t t
Reynolds Experiment dye Osborne Reynolds in 1883 Dye filament valve water outlet
Reynolds Experiment Laminar Flow Initially, at low velocities, dye filament remained intact throughout the length of the tube Turbulent Flow : Increase in velocity: velocity eventually reached at which dye: filament broke up. Colour was diffused over the whole cross-section. Particles no longer moved in an orderly manner. Occupied different relative positions in successive cross-sections. Inertia Forces versus Viscous Forces When motion of fluid particle disturbed, its inertia will tend to carry it on in the new direction. Viscous forces due to the surrounding fluid will tend to make it conform to the motion of the rest of the stream
REAL FLUIDS I. Laminar Flow Between Solid Boundaries 1. Introduction to Laminar Flow. Reynolds Experiment Reynolds Number 3. Steady Laminar Flow in Circular Pipes Hagen-Poiseuille Law 4. Steady Laminar Flow Between Parallel Planes Steady Laminar Flow Between Parallel Planes with One Plane Moving 1. Introduction to Laminar Flow The various ways of classifying flows have been met in earlier notes. Below is a summary of the broad classifications: Flow Steady/Unsteady Uniform/Non-uniform Laminar/Turbulent Description The manner in which the fluid velocity varies with time. In steady flow the velocity is unchanging at any site with time. If you measure the fluid velocity at time T=0 and you measure the velocity some time T=t 1 after, at the same place, then the velocity would be the same. In unsteady flow, the velocities would not be the same. The manner in which the fluid velocity varies in space. In uniform flow, the velocity is unchanging at any place with time. If you measure the velocity at a point O, and then you measure the velocity at another point X in the fluid, the velocities would be the same for any time at which the experiment is performed. In non-uniform flow, the velocities at the two points would be different. The manner in which fluid particles move relative to each other. In laminar flow, individual particles of fluid follow paths that do not cross those of neighbouring particles. 1
Streamlines Streamlines help in our understanding fluid flow. Recall from previous notes that a streamline is an imaginary curve drawn through a mass of fluid. The streamlines at time t form the family of curves everywhere tangent to the velocity field at that time (every point on the streamline is tangent to the net velocity vector). When the flow is steady, the fluid particles move along the streamlines. There can be no net flow across a streamline. Flow Regimes At low flow rates, fluids move in "layers". The velocity components of the individual fluid elements do not cross the streamlines. This type of behavior is called laminar flow (or streamline flow). In pressure driven flows laminar flow is predominant only at low flow rates. At higher flow rates, the streamlines are disrupted by eddies moving in all directions. These eddies make the flow turbulent, and although they cross and re-cross the streamlines, the net velocity is such that the net flow from the eddies is zero. Eddies form from contact of the fluid with a solid boundary or from two fluid layers moving at different speeds. As these eddies grow, the laminar flow becomes unstable and velocities and pressures in the flowing fluid no longer have constant or smoothly varying values. In pipes, relatively large rotational eddies form in regions of high shear near the pipe wall. These degenerate into smaller eddies as energy is dissipated by action of viscosity. The presence of eddies means that the local velocity is not the same as the bulk velocity and that there are components of velocity in all directions. The transition between laminar and turbulent flow is fuzzy. The behavior depends on entrance conditions and distance from the inlet. Often it is useful to speak of a transition region for flows that are neither laminar nor fully turbulent. Distinguishing Features of Laminar Flow As the name suggests, the flow is characterized by layers of fluid travelling at different velocities. Individual particles are held in place by molecular forces that prevent them from wandering outside their streamline.
Viscous forces predominate over inertia forces. Velocity gradients set up across the flow. Occurrences Associated with slow moving, viscous fluids. It is relatively rare in nature, but one example is flow of water through an aquifer. The velocities may be as low as a few metres per year. In lubrication between moving machinery components such as the piston in cylinder in engines. The text treats several examples that have specific interest to engineering. We will only be looking at a few of these. We look at flows in circular pipes since they are the most commonly used shape. Then we consider flows between parallel plates. An extension to this is consideration of flow characteristics when one of the plates is moving relative to the other. Of interest, but not treated here are the experiments for measurement of viscosity. Viscous Forces: Only Newtonian fluids are being considered in this course. Recall from the first few lectures that for such fluids, the shear stress developed between layers moving relative to each other was given by the following relation: du τ = µ (1) dy where τ is the shear stress parallel to the fluid motion, du/dy is the velocity gradient perpendicular in the transverse direction and μ is the dynamic viscosity. This relation is fundamental to our development of the relation. Pressure forces: Also of importance are the forces set up by pressure at the ends of the flow. Note that for steady, fully developed flow, there is no acceleration and so the total force is zero. End conditions: The considerations below always look at positions far removed from any boundaries, for, because of discontinuities, special conditions usually apply there. Recall Reynolds experiments and that observations of fully developed flow flow in which the velocity profile ceases to change in the flow direction were always made some distance away from the entry of the tube. 3
. Reynolds Experiment Reynolds Number Refer to Massey (Section 1.9.1 and 5.7) for a description of the experiment and its significance to fluid mechanics. The Reynolds Number describes the relative importance of molecular and convective transport in a flowing stream. Since molecular transport dominates in laminar flow and convective transport in turbulent flow, the Reynolds Number also serves as an indicator of the flow regime. Laminar flows in pipes typically used in engineering applications have Reynolds numbers below 000. Fully turbulent flows will usually have Reynolds numbers greater than 4000. The magnitude of the Reynolds number is independent of the system of units. Reynolds Number will be treated further in the next topic on turbulent flow in pipes. 3. Steady Laminar Flow in Circular Pipes: The Hagen-Poiseuille Equation The development was spurred by the interest in flow of blood through veins. It should be noted that the experiments used fine capillary tube, which are rigid and very much unlike the flexible walls of blood vessels. It s applicable to our purpose; it may not accurately model blood flow. The study of pipes is important : For explaining observed phenomena. It has been noted that flow in the middle of the pipe moves faster than at other sections and that there is no flow at the pipe walls; For determining what parameters control discharge through the pipe, and therefore how can these be manipulated to put to engineering use. The figure drawn in class is a longitudinal section of a straight, circular pipe of constant internal radius R. Remember that for laminar flow, the paths of individual particles do not cross. So particles that start off close to the wall of the pipe continue to move close to the 4
wall without crossing other fluid particles that start off further away from the wall. So you can imagine that there are layers of fluid moving through the pipe, like say onion skins. Consider an infinitesimal cylindrical sleeve of fluid of thickness δ r, at a distance r from the central axis. The cylinder is moving from left to right, the velocity of the fluid on the underside of the sleeve, that is at radius r, is u; the velocity of fluid just above the sleeve, that is at radius r + δr is u + δu. Recall that steady, fully developed flow is being considered, and therefore there is no net force. The two predominant forces are pressure forces at the end, and the viscous forces acting along the direction of flow: So we have, p r ( p + δp ) πr τ πrδx = 0 π () As δx approaches zero, for steady flow, the shear stress at radius r is: r dp = dx τ (3) where p, the piezometric pressure is (p+ρgh). What this says is that the shear stress varies with the position from the centre and on the differential piezometric pressure gradient with distance measured along the direction of fluid. For laminar flow, the stress is due entirely to viscous actions and so from (1) we have du r dp µ = (4) dr dx du The minus sign because on the left because is negative. dr If µ is constant (and it is since we are considering a Newtonian fluid and assuming that temperature remains constant) then integration with respect to r gives the following: r dp u = + A (5) 4µ dx and then considering the boundary condition of u=0 at the wall, that is at r=r, gives 5
( R r ) 1 dp u = (6) 4µ dx This equation, which relates the velocity with distance from the centre of the pipe, describes a parabola whose profile is shown in the figure (drawn in class). So now that there is an expression that describes how velocity varies in a circular pipe under laminar flow, it is now possible to determine an expression for discharge: Starting from consideration of an elemental ring of thickness δ r, the elemental discharge, which is the product of area and the velocity perpendicular to the area can be written as: π dp 3 ( R r ) πrδr = ( R r r ) δr 1 dp δ Q = 4 dx µ dx (7) µ To get the discharge through the entire cross-section, integrate with respect to r between 0 and R to get: R 3 ( R r r ) 0 π dp Q = dr (8) µ dx πr Q = 8µ 4 dp dx (9) Equation 9 above is known as the Hagen-Poiseuille Equation Note: Applicable to fluids that undergoes negligible change of density. -ve sign since the pressure falls in direction of flow applies only to fully developed laminar flow the equation says that in laminar flow, the velocity in the pipe is directly proportional to the drop in piezometric pressure. Developed for straight pipes. 6
Steady Laminar Flow Between Parallel Planes with One Plane Moving Consider the sketch (drawn in class) that shows an elemental strip of fluid between the two plates, at some arbitrary distance y from the bottom plate. The top plate is moving relative to the bottom plate at a velocity of V. Note that it is assumed that the element of fluid is taken from a section far removed from the ends of the boundary planes and so the flow is fully developed at the element. Derivation of an expression for discharge: The force balance is shown in class. If the fluid at the top of the element is flowing faster than that of the element s top, and if the fluid at the bottom of the element is flowing slower than that of the element s bottom, then a force balance can look like: ( p + δp) δy τδx + ( τ + δτ ) δx ρgδxδy sinθ p δy + (10) For steady, fully developed flow acceleration is zero and therefore the net force on the element is zero. Dividing by δxδy we get δτ δp = ρg sinθ (11) δy δx δh But sinθ =. So in the limit as δ y 0 δx δp δτ u = = µ δx δy y y (1) where p = (p+ρgh). Note that the pressure difference is only in the x direction. If it were in the y direction, then there would be movement perpendicular to the flow, a condition that is absence in laminar flow. So since p does not vary with y, above may be integrated with respect to y to give: δ p u y = µ A δx y + (13) 7
8 Now assuming constant µ and integrating above again: B Ay u y x p + + = µ δ δ (14) To find A and B we use the following boundary conditions: 0 B 0 y at 0 u = = = (15) c V c x p A c y at V u µ δ δ = = = (16) From above, the velocity of the fluid is: ( ) y c V cy y x p 1 u + = δ δ µ (17) An expression for the discharge Q can be found in the same way as above. Consider Equations (17) above and recall that the elemental discharge δq at y is ubδy, where b is the width of the plate (the dimension into the plane of the paper). So, c c c Vy cy y x p b ubdy Q 0 3 0 3 1 + = = µ δ δ + µ δ δ = Vc 1 c x p b Q 3 (18) From (18) above, it should be noted that flow can occur even when the pressure gradient =0, and when this occurs, the flow is known as Couette flow.
Application: The dashpot is a device for damping vibrations of machines, or rapid reciprocating motions. See the dashpot below as an example of motion of one plate relative to another. p l D c z z 1 C We will derive expression for designing the dashpot. The diagram above is a schematic representation of a dashpot, which consists of a cylinder of diameter D moving vertically within a viscous fluid contained in a cylinder of diameter D+c. Consider downward motion of the piston. This velocity is -V p, the ve sign since motion is in direction of decreasing z. 9
The width of the flowpath (equivalent to b in the parallel plate cases considered above) is: b = πd From Equation (18) above, 3 δp c Vpc Q = πd δx 1µ (19) The piston displaces fluid at the rate of Q = πd Vp 4 3 p c Vpc πd Hence Q = πd = Vp (0) l 1µ 4 Arranging this equation, 3 D p c Vp + c = (1) 6µ l u Neglecting the shear force on the piston π Dlµ since it is very small with r respect to the other forces, the force exerted on the piston by the mechanism, F, its weight, W, and the pressure forces due to the differences in pressure on the top and bottom faces of the piston. The force equation is: πd p F W = 0 () 4 assuming that there is no acceleration. Now, p = p p + ρg z z = p + ρg l (3) ( ) ( ) p1 1 1 From (), Δp is 4 p = ( F + W ) (4) πd And then substituting expression for Δp in (3) above, ( F + W ) 3 D 4 c Vp + c = gl ρ πd (5) 6µ l 10
Since c is very small in comparison with the radius D, the L.H.S of (5) becomes V D p, so (5) becomes 3 ρglπd 3 D F + W = πµ l V 3 p (6) 4 4 c Notes: The equations (5) and (6) can be applied to upwards motion by reversing the signs of F and V p For improved accuracy, replace the flowpath, b, by π(d + c) 11