Chapter 8 Steady Incompressible Flow in Pressure Conduits

Chapter 8 Steady Incompressible Flow in Pressure Conduits

Outline 8.1 Laminar Flow and turbulent flow Reynolds Experiment 8.2 Reynolds number 8.3 Hydraulic Radius 8.4 Friction Head Loss in Conduits of Constant Cross Section 8.5 Laminar flow in circular pipes 8.6 Characters of turbulent flow 8.7 Boundary Layers

Objectives Describe the appearance of laminar and turbulent flow. Compute Reynolds number and identify the type of flow. State the characteristics of laminar, turbulent and transitional flow. Define boundary layers.

Introduction In the earlier chapter, the basic equations of continuity, energy and momentum were introduced and applied to fluid flow cases where the assumption of frictionless flow (or ideal fluid flow) was made. It is now necessary to introduce concepts which enable the extension of the previous work to real fluids in which viscosity is accepted and frictional effects cannot be ignored. The concept of Reynolds number as an indication of flow type will be used extensively.

8.1 Laminar Flow and turbulent flow The flow of a fluid in a pipe may be laminar or turbulent. Osborne Reynolds first distinguished the two flows. Laminar flow: Fluid moves in smooth streamlines. Turbulent flow: Violent mixing, fluid velocity at a point varies randomly with time. Time dependence of fluid velocity at a point.

Reynolds Experiment Dye water Open to increase velocity Valve

Laminar flow Dye streak stable Low velocity Transition Turbulent flow Dye mixed up in the cross-section High velocity

Laminar: Transition: h f ν Turbulent : h f ν n 1.75 < n < 2 Point B higher critical point Point A lower critical point The point where flow transforms from laminar to turbulent is called the critical point.

Reynolds number Whether flow is laminar or turbulent depends on the dimensionless Reynolds Number (which is usually abbreviated Re) Where Re DV ρ = μ DV ν ρ = density, V = mean velocity, = D = diameter, μ=dynamic viscosity ν= kinematic viscosity ν = μ ρ

8.2 Critical Reynolds Number Point B occurs at about Re=4000,but it could be pushed to 5000, however at those valves the laminar flow is highly unstable. Point A Re=2000, below that value all flows are laminar any turbulence is damped by viscous friction. True critical Reynolds number Rec= 2000

What are the units of this Reynolds number? the Reynolds number, Re, is a non-dimensional number.

Pipe flow 2 πd A R = = 4 = χ πd d 4 d Re = ρvd μ > Re c = 2000 Turbulent flow

Open channel flow R A ( b + mh) h = = χ 2 b + 2h 1 + m b m h Re = ρ vr μ =Re > Re c = 500 Turbulent flow

Example If the pipe and the fluid have the following properties: We want to know the maximum velocity when the Re is 2000.

In laminar flow the motion of the particles of fluid is very orderly with all particles moving in straight lines parallel to the pipe walls. But what is fast or slow? And at what speed does the flow pattern change? And why might we want to know this?

What does this abstract number mean? It can be interpreted that when the inertial forces dominate over the viscous forces (when the fluid is flowing faster and Re is larger) then the flow is turbulent. When the viscous forces are dominant (slow flow, low Re) they are sufficient enough to keep all the fluid particles in line, then the flow is laminar.

] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] d d [ ] [ ] [ Re μ vl ρ T F μ Lv L v L μ y d d u Aμ T L v ρ L v L ρ x u Mu Ma F = = = = = = = = = 2 2 2 2 3 Viscous force : Inertia force

8.3 Hydraulic Radius The hydraulic radius is the flow area divided by the wetted perimeter. R h = A P For a circular pipe flowing full. 2 πr r R h = = = 2 π r 2 D 4

8.4 Friction Head Losses Steady flow in a conduit of uniform cross section A (Not necessarily circular) For equilibrium in steady flow p1 A p2 A γlasin a τ PL = 0 0 p1 p2 PL ( z 1 + ) ( z2 + ) = τ 0 = γ γ γa h f F = ma = 0 h f sin = z z L 2 1 a L γ ( A / P) τ 0 = τ 0 L γr Applies to: any shape of uniform cross section ; laminar or turbulent flow = h

Question The water in a vertical pipe flows from the top down. There are two cross-sections with the distance l, their piezometric head difference h and the frictional losses between the two cross-section h f, then: A. h f =h; B. h f =h +l; C. h f =l -h; D. h f =l.

Friction in circular conduits For circular pipes of constant cross-section 2 L V h f = f Darcy-Weisbach Equation D 2g L = pipe length D = pipe diameter V = pipe velocity f = friction factor

Friction in noncircular conduits Use D = 4R h in Darcy-Weisbach h f = f L D 2 V 2g = f L 4R h 2 V 2g With ρvd ρv. 4R Re = = μ μ

8.5 Laminar flow in circular pipes Hagen-Poiseuille law for laminar flow h f ~ V Equation involves no empirical coefficient Equation involves only fluid properties, g, and V

8.5 Laminar flow in circular pipes For laminar flow is circular pipes, the friction factor (f) is given below f 64 ν = = DV 64 Re Re = Reynolds number

Shear stress linear distribution Fig8.3 Velocity profile in laminar flow and distribution of shear stress

Shear Stress in Pipes A horizontal steady uniform flow in a circular pipe is shown in figure. Similarly: so or Physical meaning: For the uniform flow in a circular pipe, the shear stress is zero at the center of the pipe and increases linearly with the radius to a maximum τ 0 at the wall.

Velocity Distribution Laminar flow -- Newton s law of viscosity is valid: dv τ = μ dy dv dy dv dr = rγ dh = 2μ ds 2 dv dr rγ dh = 2 ds rγ dh dv = dr 2μ ds r γ dh V = + C 4μ ds 2 r = 0 γ dh r V 1 4μ ds r0 2 r V = Vmax 1 r0 r = 0 γ dh C 4μ ds 2 2 Velocity distribution in a pipe (laminar flow) is parabolic with maximum at center.

Example (8.6.2)

Question The shearing stress distribution over the cross-section of a flow in circle pipe is: A. constant over the whole cross-section; B. B. zero at the pipe axis and proportional to radius; C. zero on the pipe wall and increases linearly from the wall to the pipe axis; D. parabolic distribution.

Maximum Velocity The maximum velocity for laminar flows in a circular pipe is at the pipe center line. (r=0): u x max = ρgj 4μ r 2 0

Mean Velocity max 2 0 2 0 0 2 2 0 2 0 0 2 1 8 )2 ( 4 2 0 0 x r r x A x u r gj r rdr r r gj r rdr u A da u v = = = = = μ ρ π π μ ρ π π The mean velocity of the laminar flow in a circular pipe is half of the maximum velocity.

Question The velocity distribution of laminar flow in a circle pipe meets: A. symmetrical law B. linear change law C. parabolic law D. logarithmic curve law.

Question For the laminar flow in a circle pipe, if the velocity at pipe axis is 4m/s, then the mean velocity over the cross-section is: A. 4m/s; B. 3.2m/s; C. 2m/s; D. 1m/s.

8.6 Basic Theory of Turbulent Flow Almost all fluid flow which we encounter in daily life is turbulent. Typical examples are flow around (as well as in) cars, aeroplanes and buildings.

8.6.1 Characters of turbulent flow Irregularities randomness Deterministic approaches cannot be used (statistics)

Characters of turbulent flow (contd) Diffusivity mixing Increased momentum, heat and mass transfer

Characters of turbulent flow (contd) Energy consumption besides the energy consumption caused by viscosity, there is more energy consumption caused by turbulent shear stress.

What is the origin of turbulence? Turbulence often originates as an instability of laminar flow For example pipe flow starts to become turbulent at Re2000-2500 (the increasing Re pipe flow)

8.6.2 Fluctuation and Time-Average of Turbulent Flow Quantities In turbulent flow we usually divide the variables in one time-averaged part, which is independent of time (when the mean flow is steady), and one fluctuating part.

u x = u x u x time-average velocity u x u x 1 = T = u T 0 x u x dx u x

Decomposition of properties Consider the turbulent velocity at a point: The velocity is decomposed into a mean, u, and a fluctuating component, u

Decomposition of properties cont d Similarly for all fluid properties: We determine the time average at a fixed point as: It is important that the integral is taken over sufficiently long time interval for the average to be independent of the time overall steady state flow! If this condition is met the average of the fluctuating components is zero:

Turbulent Shear Stress Consider turbulent flow in a horizontal pipe, and the upward eddy motion of fluid particles in a layer of lower velocity to an adjacent layer of higher velocity through a differential area da Then the turbulent shear stress can be expressed as

8.6.3 Turbulent Shear Stress The turbulent shear stress consists of two parts: the laminar component, and the turbulent component, The velocity profile is approximately parabolic in laminar flow, it becomes flatter or fuller in turbulent flow. The fullness increases with the Reynolds number, and the velocity profile becomes more nearly uniform, however, that the flow speed at the wall of a stationary pipe is always zero (no-slip condition).

Turbulent Shear Stress Experimental results show that is usually a negative quantity. Terms such as or are called Reynolds stresses or turbulent stresses. Many semi-empirical formulations have been developed that model the Reynolds stress in terms of average velocity gradients. Such models are called turbulence models. Momentum transport by eddies in turbulent flows is analogous to the molecular momentum diffusion.

8.7 Boundary Layers Very thin region near solid surfaces, where flow and surface interact through viscous forces Although it accounts for a very small region in space, it is extremely important in applications involving fluid flows Boundary layers can be laminar and turbulent!!

Boundary layers

y U free stream laminar to turbulent transition edge of boundary layer x laminar turbulent δ(x) x cr Fig Schematic of boundary layer flow over a flat plate

the characteristics of boundary layer The boundary layer thickness δ grows continuously from the start of the fluid-surface contact, e.g. the leading edge. It is a function of x, not a constant. Velocity profiles and shear stress τ are f(x,y). The flow will generally be laminar starting from x = 0. The flow will undergo laminar-to-turbulent transition if the streamwise dimension is greater than a distance x cr corresponding to the location of the transition Reynolds number Re c. Outside of the boundary layer region, free stream conditions exist where velocity gradients and therefore viscous effects are typically negligible.

Reynolds number defined as where Re x = ρ U x μ = U x υ ρ = fluid density; μ = fluid dynamic viscosity ν= fluid kinematic viscosity; U = characteristic flow velocity x = characteristic flow dimension

thickness of the boundary layer We define the thickness of this boundary layer as the distance from the wall to the point where the velocity is 99% of the free stream velocity, the velocity in the middle of the pipe or river.

Plane boundary layers build-up Definition of boundary layer thickness