Monroe L. Weber-Shirk S chool of Civil and Environmental Engineering External Flows CEE 331 June 25, 2015 CEE 331 June 25, 2015 

Overview ä Non-Uniform Flow ä Boundary Layer Concepts ä Viscous Drag ä Pressure Gradients: Separation and Wakes ä Pressure Drag ä Shear and Pressure Forces ä Vortex Shedding ä Non-Uniform Flow ä Boundary Layer Concepts ä Viscous Drag ä Pressure Gradients: Separation and Wakes ä Pressure Drag ä Shear and Pressure Forces ä Vortex Shedding

Non-Uniform Flow ä In pipes and channels the velocity distribution was uniform (beyond a few pipe diameters or hydraulic radii from the entrance or any flow disturbance) ä In external flows the boundary layer is always growing and the flow is non- uniform ä In pipes and channels the velocity distribution was uniform (beyond a few pipe diameters or hydraulic radii from the entrance or any flow disturbance) ä In external flows the boundary layer is always growing and the flow is non- uniform

Boundary Layer Concepts ä Two flow regimes ä Laminar boundary layer ä Turbulent boundary layer ä with laminar sub-layer ä Calculations of ä boundary layer thickness ä Shear (as a function of location on the surface) ä Drag (by integrating the shear over the entire surface) ä Two flow regimes ä Laminar boundary layer ä Turbulent boundary layer ä with laminar sub-layer ä Calculations of ä boundary layer thickness ä Shear (as a function of location on the surface) ä Drag (by integrating the shear over the entire surface)

Flat Plate: Parallel to Flow U x y U U U   Why is shear maximum at the leading edge of the plate? boundary layer thickness shear is maximum

Laminar Boundary Layer: Shear and Drag Force Boundary Layer thickness increases with the _______ ______ of the distance from the leading edge of the plate On one side of the plate! Based on momentum and mass conservation and assumed velocity distribution square root Integrate along length of plate

Laminar Boundary Layer: Coefficient of Drag Dimensional analysis

Transition to Turbulence ä The boundary layer becomes turbulent when the Reynolds number is approximately 500,000 (based on length of the plate) ä The length scale that really controls the transition to turbulence is the _________________________ ä The boundary layer becomes turbulent when the Reynolds number is approximately 500,000 (based on length of the plate) ä The length scale that really controls the transition to turbulence is the _________________________ boundary layer thickness Re  = 3500 =

Transition to Turbulence U x y U U   U turbulent Viscous sublayer Viscous sublayer This slope (du/dy) controls  0. Transition (analogy to pipe flow)

more rapidly Turbulent Boundary Layer: (Smooth Plates) Derived from momentum conservation and assumed velocity distribution Integrate shear over plate Grows ____________ than laminar 5 x 10 5 < Re l < 10 7 x 5/4

Boundary Layer Thickness ä Water flows over a flat plate at 1 m/s. Plot the thickness of the boundary layer. How long is the laminar region? Grand Coulee x = 0.5 m

Flat Plate Drag Coefficients 1 x x x x x x x x x x 10 -6

Example: Solar Car ä Solar cars need to be as efficient as possible. They also need a large surface area for the (smooth) solar array. Estimate the power required to counteract the viscous drag at 40 mph ä Dimensions: L: 5.9 m W: 2 m H: 1 m ä Max. speed: 40 mph on solar power alone ä Solar Array: 1200 W peak ä Solar cars need to be as efficient as possible. They also need a large surface area for the (smooth) solar array. Estimate the power required to counteract the viscous drag at 40 mph ä Dimensions: L: 5.9 m W: 2 m H: 1 m ä Max. speed: 40 mph on solar power alone ä Solar Array: 1200 W peak air = 14.6 x10 -6 m 2 /s  air = 1.22 kg/m 3

Viscous Drag on Ships ä The viscous drag on ships can be calculated by assuming a flat plate with the wetted area and length of the ship Lr3Lr3 Lr3Lr3 scales with ____

Separation and Wakes ä Separation often occurs at sharp corners ä fluid can’t accelerate to go around a sharp corner ä Velocities in the Wake are ______ (relative to the free stream velocity) ä Pressure in the Wake is relatively ________ (determined by the pressure in the adjacent flow) ä Separation often occurs at sharp corners ä fluid can’t accelerate to go around a sharp corner ä Velocities in the Wake are ______ (relative to the free stream velocity) ä Pressure in the Wake is relatively ________ (determined by the pressure in the adjacent flow) small constant

Flat Plate: Streamlines U PointvC p p <U >0 >U<0 >p 0 <p 0 Points outside boundary layer!

Application of Bernoulli Equation In air pressure change due to elevation is small U = velocity of body relative to fluid

Flat Plate: Pressure Distribution CpCp 0.8 C d = 2 0 <U >U 1 2 3

Bicycle page at Princeton Drag of Blunt Bodies and Streamlined Bodies ä Drag dominated by viscous drag, the body is __________. ä Drag dominated by pressure drag, the body is _______. ä Whether the flow is viscous- drag dominated or pressure- drag dominated depends entirely on the shape of the body. ä Drag dominated by viscous drag, the body is __________. ä Drag dominated by pressure drag, the body is _______. ä Whether the flow is viscous- drag dominated or pressure- drag dominated depends entirely on the shape of the body. streamlined bluff

Velocity and Drag: Spheres Spheres only have one shape and orientation! General relationship for submerged objects Where C d is a function of Re

How fast do particles fall in dilute suspensions? ä What are the important parameters? ä Initial conditions ä After falling for some time... ä What principle or law could help us? ä What are the important parameters? ä Initial conditions ä After falling for some time... ä What principle or law could help us? Acceleration due to gravity drag Newton’s Second Law...

Sedimentation: Particle Terminal Fall Velocity

Particle Terminal Fall Velocity (continued) General equation for falling objects Relationship valid for spheres

Drag Coefficient on a Sphere Reynolds Number Drag Coefficient Stokes Law Re= Turbulent Boundary Layer

Drag Coefficient for a Sphere Equations Laminar flow R < 1 Transitional flow 1 < R < 10 4 Fully turbulent flow R > 10 4

Example Calculation of Terminal Velocity Determine the terminal settling velocity of a cryptosporidium oocyst having a diameter of 4  m and a density of 1.04 g/cm 3 in water at 15°C. Reynolds

Pressure Gradients: Separation and Wakes Van Dyke, M An Album of Fluid Motion. Stanford: Parabolic Press. Diverging streamlines

Adverse Pressure Gradients ä Increasing pressure in direction of flow ä Fluid is being decelerated ä Fluid in boundary layer has less ______ than the main flow and may be completely stopped. ä If boundary layer stops flowing then separation occurs ä Increasing pressure in direction of flow ä Fluid is being decelerated ä Fluid in boundary layer has less ______ than the main flow and may be completely stopped. ä If boundary layer stops flowing then separation occurs inertia Streamlines diverge behind object

Point of Separation ä Predicting the point of separation on smooth bodies is beyond the scope of this course. ä Expect separation to occur where streamlines are diverging (flow is slowing down) ä Separation can be expected to occur around any sharp corners ä Predicting the point of separation on smooth bodies is beyond the scope of this course. ä Expect separation to occur where streamlines are diverging (flow is slowing down) ä Separation can be expected to occur around any sharp corners (where streamlines diverge rapidly)

Drag on Immersed Bodies (more shapes) ä Figures bodies with drag coefficients on p in text. ä hemispherical shell0.38 ä hemispherical shell1.42 ä cube1.1 ä parachute1.4 ä Figures bodies with drag coefficients on p in text. ä hemispherical shell0.38 ä hemispherical shell1.42 ä cube1.1 ä parachute1.4 Why? Vs ? Velocity at separation point determines pressure in wake. The same!!!

Shear and Pressure Forces ä Shear forces ä viscous drag, frictional drag, or skin friction ä caused by shear between the fluid and the solid surface ä function of ___________and ______of object ä Pressure forces ä pressure drag or form drag ä caused by _____________from the body ä function of area normal to the flow ä Shear forces ä viscous drag, frictional drag, or skin friction ä caused by shear between the fluid and the solid surface ä function of ___________and ______of object ä Pressure forces ä pressure drag or form drag ä caused by _____________from the body ä function of area normal to the flow surface area length flow separation

Example: Matrix Power C d = 0.32 Height = m Width = m Length = m Ground clearance = 15 cm 100 kW at 6000 rpm Max speed is 124 mph Calculate the power required to overcome drag at 60 mph and 120 mph. Where does separation occur? What is the projected area?

Electric Vehicles ä Electric vehicles are designed to minimize drag. ä Typical cars have a coefficient drag of ä The EV1 has a drag coefficient of ä Electric vehicles are designed to minimize drag. ä Typical cars have a coefficient drag of ä The EV1 has a drag coefficient of Smooth connection to windshield

Drag on a Golf Ball DRAG ON A GOLF BALL comes mainly from pressure drag. The only practical way of reducing pressure drag is to design the ball so that the point of separation moves back further on the ball. The golf ball's dimples increase the turbulence in the boundary layer, increase the _______ of the boundary layer, and delay the onset of separation. The effect is plotted in the chart, which shows that for Reynolds numbers achievable by hitting the ball with a club, the coefficient of drag is much lower for the dimpled ball. inertia Why not use this for aircraft or cars?

Effect of Turbulence Levels on Drag ä Flow over a sphere: (a) Reynolds number = 15,000; (b) Reynolds number = 30,000, with trip wire. Point of separation Causes boundary layer to become turbulent

Effect of Boundary Layer Transition Ideal (non viscous) fluid Real (viscous) fluid: laminar boundary layer Real (viscous) fluid: turbulent boundary layer No shear! Increased inertia in boundary layer

Spinning Spheres ä What happens to the separation points if we start spinning the sphere? LIFT!

Vortex Shedding ä Vortices are shed alternately from each side of a cylinder ä The separation point and thus the resultant drag force oscillates ä Frequency of shedding (n) given by Strouhal number S ä S is approximately 0.2 over a wide range of Reynolds numbers ( ,000,000) ä Vortices are shed alternately from each side of a cylinder ä The separation point and thus the resultant drag force oscillates ä Frequency of shedding (n) given by Strouhal number S ä S is approximately 0.2 over a wide range of Reynolds numbers ( ,000,000)

Summary: External Flows ä Spatially varying flows ä boundary layer growth ä Example: Spillways ä Two sources of drag ä shear (surface area of object) ä pressure (projected area of object) ä Separation and Wakes ä Interaction of viscous drag and adverse pressure gradient ä Spatially varying flows ä boundary layer growth ä Example: Spillways ä Two sources of drag ä shear (surface area of object) ä pressure (projected area of object) ä Separation and Wakes ä Interaction of viscous drag and adverse pressure gradient

Solution: Solar Car U = m/s l = 5.9 m air = 14.6 x m 2 /s Re l = 7.2 x 10 6 C d = 3 x  air = 1.22 kg/m 3 A = 5.9 m x 2 m = 11.8 m 2 F d =14 N P =F*U=250 W

Reynolds Number Check R<<1 and therefore in Stokes Law range R = 1.1 x 10 -6

Solution: Power a Toyota Matrix at 60 or 120 mph P = 9.3 kW at 60 mph P = 74 kW at 120 mph

Grand Coulee Dam Turbulent boundary layer reaches surface!

Drexel SunDragon IV ä Vehicle ID: SunDragon IV (# 76) Dimensions: L: 19.2 ft. (5.9 m) W: 6.6 ft. (2 m) H: 3.3 ft. (1 m) Weight: 550 lbs. (249 kg) Solar Array: 1200 W peak; 8 square meters terrestrial grade solar cells; manf: ASE Americas Batteries: 6.2 kW capacity lead-acid batteries; manf: US Battery Motor: 10 hp (7.5 kW) brushless DC; manf: Unique Mobility Range: Approximately 200 miles (at 35 mph on batteries alone) Max. speed: 40 mph on solar power alone, 80 mph on solar and battery power. Chassis: Graphite monocoque (Carbon fiber, Kevlar, structural glass, Nomex) Wheels: Three 26 in (66 cm) mountain bike, custom hubs Brakes: Hydraulic disc brakes, regenerative braking (motor)

Pressure Coefficients on a Wing NACA Flowfield Cp's

Shear and Pressure Forces: Horizontal and Vertical Components lift drag U Parallel to the approach velocity Normal to the approach velocity  A defined as projected area _______ to force! normal drag lift p < p 0 negative pressure p < p 0 negative pressure p > p 0 positive pressure