OUTLINE TACOMA NARROWS BRIDGE FLOW REGIME PAST A CYLINDER VORTEX SHEDDING MODES OF VORTEX SHEDDING PARALLEL & OBLIQUE FLOW PAST A SPHERE AND A CUBE SUMMARY
TACOMA NARROWS BRIDGE, USA THE BRIDGE COLLAPSED IN NOVEMBER 1940 AFTER 4 MONTHS OF ITS OPENING TO TRAFFIC!
TACOMA NARROWS BRIDGE, USA The dramatic collapse was induced due to VORTEX SHEDDING AT A WIND SPEED OF 42MPH! This is how strong a flow regime past a bluff body can become.
OVERVIEW OF FLOW REGIME PAST A CYLINDER The transitions in the flow regimes is affected by the roughness, turbulence levels, cylinder aspect ratio, end conditions and blockage. Re =40-150 STABLE LAMINAR SHEDDING REGIME. Re= 150-300 TRANSITION REGIME. Re= 300-10000+ IRREGULAR REGIME.
Flow past a cylinder (a) The flow past a cylinder for a very low Reynolds number. The flow smoothly divides and reunites around the cylinder. (b) At a Reynolds number of about 4, the flow (boundary layer) separates in the downstream and the wake is formed by two symmetric eddies. The eddies remain steady and symmetrical but grow in size up to a Reynolds number of about 40. (c) At a Reynolds number around 47, the wake starts shedding vortices into the stream.
VORTEX SHEDDING At around Re=47 vortex shedding begin to take place. A Hopf bifurcation is breaks the symmetry between the two eddies and with a slight bifurcation the vortex detaches itself from the body. In its place a new vortex is formed. The airflow past the object creates alternating low-pressure vortices on the downwind side of the object. The object will tend to move toward the low-pressure zone.
VON KÁRMÁN VORTEX STREET Over a large Re range (47<Re<10 7 for circular cylinders), eddies are shed continuously from each side of the body, forming rows of vortices in its wake. A Von Kármán vortex street is a repeating pattern of swirling vortices caused by the unsteady separation of flow over bluff bodies. They are named after the engineer & fluid dynamicist, Theodore von Kármán.
VON KÁRMÁN VORTEX STREET A video showing a smoke visualization of the FORMATION OF VORTEX STREETS as wind flows past a cylindrical body.
FORMATION OF VORTEX STREETS Vortices are shed in to the downstream flow from alternate sides of the body ( with alternate senses of rotation ), giving the appearance of alternately opposite signed vortices. Karman investigated the phenomenon and concluded that a nonstaggered row of vortices is unstable, and a staggered row is stable only if the ratio of lateral distance between the vortices to their longitudinal distance is 0.28. Because of the similarity of the wake with footprints in a street, the staggered row of vortices behind a blue body is called a Karman Vortex Street. The vortices move downstream at a speed smaller than the upstream velocity U.
VORTEX DISLOCATIONS: Origin of fluctuations Existence of vortex dislocations in wake transition. Primary vortices move out of phase. Irregularities dominates downstream. LARGE LOW FREQUENCY IRREGULARITIES: Cross sectional and plane view
STROUHAL NUMBER When considering a long circular cylinder, the frequency of vortex shedding is given by the empirical formula Strouhal number= fd/v where f = Vortex shedding frequency While an eddy on one side is shed, that on the other side forms, resulting in an unsteady flow near the cylinder. As vortices of opposite circulations are shed off alternately from the two sides, the circulation around the cylinder changes sign, resulting in a lateral force. If the frequency of vortex shedding is close to the natural frequency of some mode of vibration of the cylinder body, then an appreciable lateral vibration culminates. This may cause large fluctuating pressure forces leading to structural vibrations, acoustic noise, or resonance.
EFFECTS OF VORTEX SHEDDING When the frequency of vortex shedding matches the resonance frequency of the structure, the structure will begin to resonate and the structure's movement can become self-sustaining. Vortex shedding was one of the causes proposed for the failure of the Tacoma Narrows Bridge in 1940 A thrill ride "Vertigo" in Ohio suffered the vortex shedding during the winter of 2001, one of the three towers collapsed. Vortex shedding caused the collapse of three towers at Ferrybridge power station in 1968 during high winds.
Flow at higher Reynolds number At about Re = 500, multiple frequencies start showing up and the wake tends to become Chaotic. As the Reynolds number becomes higher, the boundary layer around the cylinder tends to become turbulent. The wake, shows fully turbulent characters. For larger Reynolds numbers, the boundary layer becomes turbulent. A turbulent boundary layer offers greater resistance to separation than a laminar boundary layer. As a consequence the separation point moves downstream and the separation angle is delayed to 110 0 from the forward stagnation point
STROUHAL NUMBER vs. REYNOLDS NUMBER Discontinuity is observed in the Strouhal (S) vs. Reynolds no. (Re) relationship in the laminar shedding regime. Scatter of the order of 20% Disparity, though U, D, f can be measured to 1% accuracy. Many explanations have been given over the years. S-Re discontinuity is explained by a changeover from one mode of oblique shedding to another. S = fd/v Re=ρVD/μ
MODES OF VORTEX SHEDDING PARALLEL MODE OBLIQUE MODE
SQUIRE S TRANSFORMATION For a given wake profile and Reynolds number, and for a parallel flow, if f is the frequency & σ is temporal growth of the most unstable 2D wave, then for an oblique wave at an angle θ f(θ)= f cos θ & σ(θ)= σ cos θ
UNIVERSAL STROUHAL CURVE With parallel shedding, the Strouhal - Reynolds Number curve is completely continuous. Garry Brown suggested that cylinder wake frequencies follow the same trend as oblique waves. Williamson defined a universal Strouhal curve in which the experimental oblique shedding data [S θ ] closely collapses onto the parallel shedding curve [S o ] by the transformation : S o = S θ / cosθ
OBLIQUE MODE OF VORTEX SHEDDING Even for a cylinder that is hundreds of diameters in length, the angle of shedding depends on the particular boundary conditions at the span wise ends of the cylinder. Shedding assumes a cellular structure, with different frequencies co-existing at different span wise locations.
PARALLEL MODE OF VORTEX SHEDDING Promotion of oblique shedding to parallel vortex shedding: Very large aspect ratios ( L/D > 2000) Slight speeding up of the flow near the ends
Techniques to manipulate end boundaries USING EDGE OF END PLATE: angling inwards the leading edge of endplates. USING COAXIAL END CYLINDERS: Ending the span with larger coaxial cylinders
Techniques to manipulate end boundaries USING CONTROL CYLINDERS: locating large cylinders normal and upstream of the test cylinder. USING SUCTION TUBES FROM DOWNSTREAM: incident flow near the ends is speeded up by suction pipe.
Flow past spheres: MAGNUS EFFECT A Transverse aerodynamic force can be detected on a rotating body. Magnus force results from the asymmetric distortion of the boundary layer displacement thickness caused by the combined spinning and flow past the sphere. Boundary layer separation is delayed on the side of the spinning object that is moving in the same direction as the free stream flow, while the separation occurs prematurely on the side moving against the free stream flow. The wake then shifts toward the side moving against the free stream flow.
AERODYNAMICS OF CRICKET BALLS The key to making a cricket ball swing is to cause a pressure difference between the two sides of the ball. Swing is achieved by keeping one side of the ball polished and bowling while keeping the seam at particular angle. This turbulent boundary layer by virtue of its increased energy, separates relatively late compared to the boundary layer on the nonseam side.
FLOW PAST SPHERES Calculated streamlines colored by velocity at Re = 100. Flow is steady and axis symmetric and exhibits a large (~D) toroidal vortex in the near wake. By Re = 211 the axial symmetry of the flow past the sphere breaks down although the flow remains temporarily steady. This figure shows select streamlines at Re = 250.
By Re = 270, the flow field becomes unsteady but periodic. Streamlines colored by pressure are shown here for every quarter period at Re = 300. The unsteadiness results from the rapid growth of a portion of the wake vortex. The section of the vortex formed in the separating shear layer at t = T/2 quickly outgrows its equilibrium strength. As it increases in strength, it eventually cuts itself from the wake and sheds.
FLOW PAST A CUBE Unlike spherical cylinders and cube, the square cylinders have sharp corners. The formation of vortices depends on the edges and the corners. The simulation of an external flow around a square cylinder at Re=22,000
SUMMARY Vortex shedding, vortex dislocations and different vortex shedding regimes. Oblique shedding and influence of end boundary conditions. Variation of Strouhal number with Reynolds no. Vortex shedding around spheres and cubes.
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