Associate Professor, IIT (BHU), Varanasi Flow control past bluff bodies Dr. Om Prakash Singh Associate Professor, IIT (BHU), Varanasi www.omprakashsingh.com

What is a Bluff body

Inviscid vs. Viscous flows Theoretical: Beautifully behaved but mythically thin boundary layer and wake region Actual: High separated Flow and large wake region NO DRAG HIGH DRAG

COMPARISON OF DRAG FORCES

GOLF BALL AERODYNAMICS Large Wake of Separated Flow, High Pressure Drag Laminar B.L. Separation Point Reduced Size Wake of Separated Flow, Lower Pressure Drag Turbulent B.L. Separation Point

GOLF BALL AERODYNAMICS Laminar B.L. Turbulent B.L. Laminar B.L. Turbulent B.L. Pressure drag dominates sphere Dimples encourage formation of turbulent B.L. Turbulent B.L. less susceptible to separation Delayed separation → Less drag

Basics of External Flows Blunt shape Streamline shape Dimples reduced wake region, fluid remain attached to the surface due to turbulent Boundary layer

Golf ball Boundary layer separation on a sphere for (A) laminar flow and (B) turbulent flow caused by roughing the nose. Note separation occurs much downsteam in turbulent flow. This is the reason dimples are provided on the golf ball. The ball experiences less drag and hence travels long distance.

Basics of External Flows The details of a flow around a blunt body.

Why is the study of bluff body flows important ? Opened to traffic on July 1, 1940 Collapsed on Nov. 7, 1940 The bridge was the third-longest suspension bridge in the world The bridge's collapse had a lasting effect on science and engineering. Its failure also boosted research in the field of bridge aerodynamics-aeroelastics, the study of which has influenced the designs of all the world's great long-span bridges built since 1940.

WHY IS THE STUDY OF BLUFF BODY FLOWS IMPORTANT ? November 7, 1940

Vortex formation in nature Cloud formation pattern behind Selkirk island taken in Sept. 1999 By Landsat 7 satellite. http://www.fluids.eng.vt.edu/msc/gallery/vortex/selkirk.htm

Motivation 1. Flow Induced Vibrations 2. Acoustic noise generation 3. Fluid mixing 4 Flow control .. .Many more

One of the simplest bluff body is a circular cylinder

The Cylinder in Cross Flow Conditions depend on special features of boundary layer development, including onset at a stagnation point and separation, as well as transition to turbulence. Stagnation point: Location of zero velocity and maximum pressure. Followed by boundary layer development under a favorable pressure gradient and hence acceleration of the free stream flow . As the rear of the cylinder is approached, the pressure must begin to increase. Hence, there is a minimum in the pressure distribution, p(x), after which boundary layer development occurs under the influence of an adverse pressure gradient

Flow over cylinder Separation occurs when the velocity gradient reduces to zero. and is accompanied by flow reversal and a downstream wake.

Flow over cylinder Location of separation depends on boundary layer transition.

Effect of Pressure Gradient: p/x

Reason behind flow separation: effect of p/x Boundary layer P(x,y) U 𝜕𝑃 𝜕𝑦 ≈0 x 1>2 High P Low P 𝜕𝑃 𝜕𝑥 <0 U 2 x Boundary layer velocity profile becomes steep when flow is driven by high pressure difference in favorable condition i.e. pressure decreases in the direction of flow.

Reason behind flow separation: effect of p/x 𝜕𝑃 𝜕𝑥 >0 Low P High P U (x) x Flow direction reverse near the wall when pressure on right side is high i.e. dp/dx > 0 i.e adverse (“uphill”)condition. Such flow is called reversed flow. Boundary layer thickness increases in the direction of flow.

Reason behind flow separation: effect of p/x Velocity profiles across boundary layers with favorable and adverse pressure gradients X-momentum equation Near the wall, inertia forces becomes negligible

Reason behind flow separation: effect of p/x 𝜇 𝜕2𝑢 𝜕𝑦2 wall = 𝜕𝑃 𝑑𝑥 Flow accelerate in front of the cylinder, hence, 𝜕𝑃 𝑑𝑥 <0 i.e. pressure decrease along the flow and hence, 𝜇 𝜕2𝑢 𝜕𝑦2 wall <0 (accelerating flow) And when pressure gradients becomes positive i.e. pressure increases along the flow, 𝜇 𝜕2𝑢 𝜕𝑦2 wall >0 (decelerating flow) Clearly, in the boundary layer flow, there is a point when 𝜇 𝜕2𝑢 𝜕𝑦2 wall =0 (point of inflection)

Meaning of Second derivative A plot of f ( x ) = sin ⁡ ( 2 x ) f(x)=sin(2x) from − π / 4 - to 5 π / 4. The tangent line is blue where the curve is concave up, green where the curve is concave down, and red at the inflection points (0, π/2, and π).

Meaning of First derivative The first derivative of the function f(x), which we write as df/dx, is the slope of the tangent line to the function at the point x To put this in non-graphical terms, the first derivative tells us how whether a function is increasing or decreasing, and by how much it is increasing or decreasing. This information is reflected in the graph of a function by the slope of the tangent line to a point on the graph, which is sometimes describe as the slope of the function. Positive slope tells us that, as x increases, f(x) also increases. Negative slope tells us that, as x increases, f(x) ) decreases. Zero slope does not tell us anything in particular: the function may be increasing, decreasing, or at a local maximum or a local minimum at that point

Meaning of Second derivative The second derivative of a function is the derivative of the derivative of that function. While the first derivative can tell us if the function is increasing or decreasing, the second derivative tells us if the first derivative is increasing or decreasing. If the second derivative is positive, then the first derivative is increasing, so that the slope of the tangent line to the function is increasing as x increases. We see this phenomenon graphically as the curve of the graph being concave up, that is, shaped like a parabola open upward. Likewise, if the second derivative is negative, then the first derivative is decreasing, so that the slope of the tangent line to the function is decreasing as x increases. Graphically, we see this as the curve of the graph being concave down, that is, shaped like a parabola open downward. At the points where the second derivative is zero, we do not learn anything about the shape of the graph: it may be concave up or concave down, or it may be changing from concave up to concave down or changing from concave down to concave up

Inflection points If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa. A point where this occurs is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection.

Why boundary layer thickness increases along the flow? From continuity equation, 𝑣 𝑦 =− 0 𝑦 𝜕𝑢 𝜕𝑥 Compared to a flat plate, a decelerating externals stream causes a larger -u/x within the boundary layer because the deceleration of the outer flow adds to the viscous deceleration within the boundary layer. It follows from the above equation that the u-field, directed away from the surface, is larger for a decelerating flow. The boundary layer therefore thickens not only by viscous diffusion but also by advection away from the surface, resulting in a rapid increase in the boundary layer thickness with x. In other words, when the fluid looses its momentum in flow direction, to conserve the momentum and mass, flow expands in y-direction. Hence, increase in y-momentum causes flow to separate from the surface.

Flow separation The existence of the point of inflection implies a slowing down of the region next to the wall, a consequence of the uphill pressure gradient. Under a strong enough adverse pressure gradient, the flow next to the wall reverses direction, resulting in a region of backward flow. The reversed flow meets the forward flow at some point S at which the fluid near the surface is transported out into the mainstream. We say that the flow separates from the wall. The separation point S is defined as the boundary between the forward flow and backward flow of the fluid near the wall, where the stress vanishes (i.e.. =u/y=0): 𝜕𝑢 𝜕𝑦 wall =0 It is apparent from the figure that one streamline intersects the wall at a definite angle at the point of separation.

Important note on boundary layer separation The boundary layer equations are valid only as far as down stream at the point of separation. Beyond it the boundary layer becomes so thick that the basic underlying assumption become invalid i.e. /L << 1 becomes invalid. Moreover, the parabolic character of the boundary layer equations requires that a numerical integration is possible only in the direction of advection (along which information is propagated), which is upstream within the reversed flow region. A forward (downstream) integration of the boundary layer equation, therefore breaks down after the separation point. Last, we can no longer apply potential theory to find the pressure distribution in the separated region, as the effective boundary or the irrigational flow is no longer the solid surface but some unknown shape encompassing part of the body plus the separated region

Vorticity formation Some regimes of flow over a circular cylinder.

Vorticity formation Spiral blades used for breaking up the spanwise coherence of vortex shedding from a cylindrical rod. 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 an oscillating “lift” or 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 has been observed to result. Engineering structures such as suspension bridges and oil drilling platforms are designed so as to break up a coherent shedding of vortices from cylindrical structures. This is done by including spiral blades protruding out of the cylinder surface, which break up the span wise coherence of vortex shedding, forcing the vortices to detach at different times along the length of these structures

Vorticity formation The passage of regular vortices causes velocity measurements in the wake to have a dominant periodicity. The frequency f is expressed as a nondimensional parameter known as the Strouhal number, defined as 𝑆= 𝑓𝐷 𝑈 ∞ Experiments show that for a circular cylinder the value of S remains close to 0.21. for a large range of Reynolds numbers. For small values of cylinder diameter and moderate values of free stream velocity U, the resulting frequencies of the vortex shedding and oscillating lift lie in the acoustic range. For example, at U = 10m/s and a wire diameter of D= 2mm, the frequency corresponding to a Strouhal number of 0.21 is n = 1050 cyclcs per second. The “singing” of telephone and transmission lines has been attributed to this phenomenon

Swing of a Cricket ball The presence of the seam is able to trip the laminar boundary Layer into turbulence on one side of the ball, while the boundary layer on the other side remains laminar. Differential drag force causes ball to swing.

Photograph of a cricket ball in a wind tunnel experiment, clearly shows the delayed separation on the seam side. Note that the wake has been deflected upward by the presence of the ball, implying that an upward force has been exerted by the ball on the fluid. It follows that a downward force has been exerted by the fluid on the ball.

FLOW PAST A ROTATING CYLINDER

Vortices disappear at certain rotation

End