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