Fluid Mechanics Ch1 ptech

Cause of Shearing forces

shearing forces within a body of fluid arise due to adjacent 'layers' of fluid moving past each other

Shear Stress definition

τ=F/A

Newton's Law of Viscosity

τ=μ(du/dy) shear stress=dynamic viscosity*(change in speed/change in height of fluid element)

Flow past a bluff body Physical Mechanisms

1) Free stream flow, unaffected in character by the body 2) Boundary layer Flow: immediately adjacent to the surface of the body 3) Separated regions: the WAKE, where the boundary layer has separated from the surface of the body at the SEPARATION POINT...boundary layer slows continuously until separation point

Main Causes of Energy Loss in Pipe Flows

1) Separating Flows 2) Expanding Flows

Flow around a sharp bend physical mechanisms

1) Separation and recirculation at outside corner in area outside of main curved flow path 2) Separation point at the inside corner where horizontal flow continues outwards and starts bending down towards the vertical section 3) Vena contracta and associated pressure drop/energy losses directly after the bend where layers have contracted upon themselves from the continued straight velocity before turning 4) Recirculation directly around the inside corner below the main curved flow path

Flow Along a Flat Plate Mechanisms

1) Shear layer is created due to skin friction drag 2) Fluid slows near a wall leading to shearing between layers due to viscous friction 3) Viscosity dominant in boundary layer 4) No-slip at wall, zero relative velocity 5) Boundary layer thickness = distance away from wall at which u=0.99umax 6) Boundary layer thickness continually increases as flow continues along the flat plate, but at a decreasing rate

Fluid Definition

A fluid is a substance which deforms continuously under the action of shearing forces, however small >> for a fluid to be at rest, there must be NO shearing forces

Re vs Viscous forces

At low Re, viscous forces dominate At high Re, viscous forces unimportant

Flow Past a Bluff Body flow regimes

Bluff=not streamlined, i.e the flow is significantly affected while passing it 1) Very low Re <0.5, Slow flow, small cylinder, viscous forces dominate as fluid creeps past body 2)1<Re<10s, Beginings of flow separation, but recirculating areas are trapped 3) 250<Re<10^4, Recirculation areas RELEASED, unsteady oscillating 4) 10^4<Re<2x10^5, Laminar, separated, the wake loses its structure but separation points in same place 5) Re>2x10^5, Separation point moves much further downstream on the cylinder, boundary layer flow becomes turbulent and higher speed flows penetrate into boundary, leading to smoother overall flow.

Flow Along a Flat Plate Laminar vs Turbulent and Boundary Layer thickness equations

Boundary layer flow could be laminar or turbulent depending on Reynolds number, which is calculated here using the downstream distance X and not diameter Rex=(ρuX)/(μ) Re<10^5 Laminar 10^5 < Re < 2x10^5 Transition Re > 2x10^5 Turbulent Boundary layer thickness grows differently in laminar and turbulent LAMINAR: δ=(5X)/(Rex)^0.5 TURBULENT: δ=(0.37X)/(Rex)^0.2

Shear Strain definition

Deformation in response to the shear stress

Form Drag

Drag force is a force on a body in the downstream direction (force opposing the motion of a body flowing through a fluid) Flow past a bluff body results in form drag Skin friction drag also exists, arising due to the no-slip condition

Skin Friction Drag Formulas, Laminar vs Turbulent

F=0.5*Cf*ρ*A*u^2 A=area of contact between fluid and the plate Cf varies depending on whether the flow is laminar or turbulent LAMINAR: Cf=(1.4)/(Rex)^0.5 TURBULENT: Cf=(0.074)/(Rex)^0.2

Drag Force equation and influences

Fd=0.5Cd*ρ*A*u^2 Cd is drag coefficient and depends on the geometry of the body A is projected frontal area Varies linearly with Area and much faster with flow speed

Incompressible flows

Flows in which variations in density are negligible, including: ALL LIQUIDS Gas flows in which flow speed u<<sound speed. If less than Mach 0.3 (100m/s), incompressible. Compressible flows have large variations in density

Newtonian Fluids

Fluids obeying Newton's Law of Viscosity

Liquids vs Solids stress/strain response

For a solid, a given shear stress results in a fixed shear strain, whereas for a liquid, a given shear stress results in a fixed RATE OF CHANGE of strain

Wake Oscillation and Vortex Shedding plus equation, Strouhal Number

For long, slender bodies placed across a flow, wake is observed to OSCILLATE with alternate vortex shedding occurring from top to bottom of body, leading to LIFT FORCES at a fixed frequency. (fD/u)=0.198(1-(19.7/Re))=0.2

Dimensional analysis motivation and example

If an equation or relationship between parameters is to make sense, the dimensions of the terms must be the same, i.e. there must be the same units and dimensions to the same powers on each side of the equation, which can be checked. Example with drag force: Fd=f(u,D,ρ) >> Drag force is a function of flow speed, object dimension, and density of the fluid [Fd]=[u^a*D^b*ρ^c] [Fd]=[u]^a*[D]^b*[ρ]^c [Fd]=[force]=[mass x acceleration]=[mass x length per time per time]=MLT^-2 [u]=[speed]=[length per unit time]=LT^-1 [D]=[length]=L [ρ]=[mass per unit volume]=ML^-3 Equating dimensions: MLT^-2=(LT^-1)^a*(L)^b*(ML^-3)^c =(L^a)(T^-a)(L^b)(M^c)(L^-3c) Equate powers left n right Mass: 1=c Length: 1=a+b-3c Time: -2=-a Thus, a=2, b=2, c=1 So, Fd proportional to ρ*D^2*u^2=ρAu^2

Ideal Flow

Incompressible flow with no viscous effects and a flt velocity profile (all fluid flows at the same constant velocity

Surface tension definition and response to temperature

Inter-molecular forces usually balanced in a liquid, but at an interface, liquid surface acts like an ELASTIC MEMBRANE. Surface tension is constant over the surface of separation As temperature increases, surface tension decreases

Behavior of fluid flow near a wall/boundary

No slip condition at the boundary, zero velocity, with subsequent layers rising in velocity until reaching peak velocity in the middle of flow. Layers subsequently drag the next closest layers with decreasing effectiveness and creating a velocity gradient. Fork in yogurt example

Flow in an enclosed pipe

No slip condition at the walls Flow slows near to the wall due to viscous friction. Region affected by the presence of the wall is the BOUNDARY LAYER Viscosity effects dominant in the boundary layer Boundary layer thickness INCREASES as viscosity increases, and DECREASES as free stream flow speed increases. Variation in flow speed with cross-duct position is known as the VELOCITY PROFILE

Orifice Flow Physical Mechanisms

Orifice is a location of RESTRICTED flow 1) Stagnant region directly in front of the restricted area with little or no flow 2) Separation at the lip of the orifice, where flow continues inwards even after pipe opens up again 3) Inward flow just after orifice >> flows can't turn sharp corners so keep going inwards a bit (cause of separation) 4) Vena Contracta in the jet slightly beyond orifice (Area Jet<Orifice Area) 5) Recirculation outside jet circling back towards the orifice 6) PRESSURE DROP across orifice leading to energy losses

Turbulent Flow

Random, 3D motion, often in addition to a mean-velocity flow similar to laminar. Dye injected appears as myriad of entangled threads

Reynolds Number Definition and flow regimes

Re=(ρuD)/(μ) Re < 2000: Laminar flow 2000 < Re < 4000: Transitional (could be either laminar or turbulent, NOT in between) Re > 4000: Turbulent

Laminar flow

Smooth fluid motion in 'layers' Dye injected appears as a single line velocity varies with height but direction is constant

What does viscosity measure?

Viscosity is a measure of how easy/difficult it is to shear a fluid, i.e how easily it will flow under given shearing forces

Viscosity in liquid vs gases and with changes in temperature

Viscosity of gases is much LESS than in fluids In gases, as temperature increases, viscosity INCREASES In liquids, as temperature increases, viscosity DECREASES