All that begins... peace be upon you

All that begins... peace be upon you

Faculty of Mechanical Engineering Department of Thermo Fluids SKMM 2323 Mechanics of Fluids 2 «An excerpt (mostly) from White (2011)» ibn Abdullah May 2017

Outline 1 Rankine Oval Flow Past a Circular Cylinder with Circulation The Kutta-Joukowski Lift Theorem ibn.abdullah@dev.null c b n a 2017 SKMM 2323 Mechanics of Fluids 2 Potential Flow 1 / 11

Rankine Oval A Rankine oval is formed by a source sink pair aligned parallel to a uniform stream, and it is long compared with its height. The combined stream function is ψ = U y m tan 1 2ay = U r sin θ + x 2 + y 2 m(θ1 θ2) (1) a2 When streamlines of constant ψ are plotted from Eq.(1), an oval body shape appears, as in Figure 1. Figure 1: Rankine oval (White, 2011). ibn.abdullah@dev.null c b n a 2017 SKMM 2323 Mechanics of Fluids 2 Potential Flow 2 / 11

Rankine Oval The half-length L and half-height h of this oval depend on the relative strength of source and stream, i.e. the ratio m/(u a), which equals 1.0. There are stagnation points at the front and rear, x = ±L, and points of maximum velocity and minimum pressure at the shoulders, y = ±h, of the oval. These parameters are a function of the basic dimensionless parameter m/(u a) h a = cot h/a (2a) 2m/(U a) ( L a = 1 + 2m ) 1/2 (2b) U a and u max U = 1 + 2m/(U a) 1 + h 2 /a 2 (2c) All these parameters are shown tabulated in Figure 2. ibn.abdullah@dev.null c b n a 2017 SKMM 2323 Mechanics of Fluids 2 Potential Flow 3 / 11

Rankine Oval As we increase m/(u a) from zero to large values, the oval shape increases in size and thickness from a flat plate of length 2a to a huge, nearly circular cylinder. Figure 2: Rankine oval parameters (White, 2011). In the limit as m/(u a), L/h 1.0 and u max/u 2.0, which is equivalent to flow past a circular cylinder. ibn.abdullah@dev.null c b n a 2017 SKMM 2323 Mechanics of Fluids 2 Potential Flow 4 / 11

Flow Past a Circular Cylinder with Circulation At large source strength the Rankine oval becomes a large circle, much greater in diameter than the source sink spacing 2a. Viewed on the scale of the cylinder, this is equivalent to a uniform stream plus a doublet. If we also throw in a vortex of strength K at the doublet center, it will not change the shape of the cylinder. The stream function for flow past a circular cylinder with circulation, centred at the origin, is a uniform stream plus a doublet plus a vortex: ψ = U r sin θ λ sin θ r K ln r + constant (3) The doublet strength λ has units of velocity times length squared. For convenience, let λ = U a 2, where a is a length, and let the arbitrary constant equal K ln a. Then the stream function becomes ψ = U sin θ (r a2 r ) K ln r a (4) ibn.abdullah@dev.null c b n a 2017 SKMM 2323 Mechanics of Fluids 2 Potential Flow 5 / 11

Flow Past a Circular Cylinder with Circulation Figure 3 shows streamlines plotted for four different values of the dimensionless vortex strength K/(U a). For all cases the line ψ = 0 corresponds to the circle r = a, i.e. the shape of the cylindrical body. Figure 3: Flow past a circular cylinder with circulation for values of K/(U a) of (a) 0, (b) 1.0, (c) 2.0 and (d) 3.0 (White, 2011). ibn.abdullah@dev.null c b n a 2017 SKMM 2323 Mechanics of Fluids 2 Potential Flow 6 / 11

The Kutta-Joukowski Lift Theorem The cylinder flows with circulation, Figure 3, develop an inviscid downward lift normal to the free stream, called the Magnus-Robins force, which is proportional to stream velocity and vortex strength. We see from the streamline patterns that the velocity on top of the cylinder is less, and, thus, from Bernoulli s equation, the pressure is higher. On the bottom, we see tightly packed streamlines, high velocity, and low pressure; viscosity is neglected. Inviscid theory predicts this force. From Bernoulli s equation, neglecting gravity, the surface pressure p s is given by p + 1 2 ρu = ps + 1 2 ρ ( 2U sin θ + K a ) 2 (5) or p s = p + 1 2 ρu2 (1 4 sin 2 θ + 4β sin θ β 2 ) (6) where β = K/(U a) and p is the free-stream pressure. ibn.abdullah@dev.null c b n a 2017 SKMM 2323 Mechanics of Fluids 2 Potential Flow 7 / 11

The Kutta-Joukowski Lift Theorem If b is the cylinder depth into the paper, the drag D is the integral over the surface of the horizontal component of pressure force: D = 2π 0 (p s p ) cos θ ba dθ But the integral of cos θ times any power of sin θ over a full cycle is identically ZERO!. So, D(cylinder with circulation) = 0 (7) ibn.abdullah@dev.null c b n a 2017 SKMM 2323 Mechanics of Fluids 2 Potential Flow 8 / 11

The Kutta-Joukowski Lift Theorem The lift force L normal to the stream, taken positive upward, is given by summation of vertical pressure forces: L = 2π 0 (p s p ) sin θ ba dθ Since the integral over 2π of any odd power of sin θ is zero, only the third term in the parentheses in Eq.(6) contributes to the lift: or L = 1 4K 2 ρu2 ba au L b = ρu Γ 2π 0 sin 2 θ dθ = ρu (2πK)b and circulation Γ depends on body size and orientation through a physical requirement. Notice that L is independent of radius a of the cylinder, and the problem in airfoil analysis is reduced to determine the Γ as a function of airfoil shape and orientation. (8) ibn.abdullah@dev.null c b n a 2017 SKMM 2323 Mechanics of Fluids 2 Potential Flow 9 / 11

The Kutta-Joukowski Lift Theorem Eq.(8) was generalized by W. M. Kutta (1902) and independently by N. Joukowski (1906) as follows: According to inviscid theory, the lift per unit depth of any cylinder of any shape immersed in a uniform stream equals ρu Γ, where Γ is the total net circulation contained within the body shape. The direction of the lift is 90 from the stream direction, rotating opposite to the circulation. Homeworks: Read Section 8.7 Airfoil Theory, pp562 566 (White, 2011). ibn.abdullah@dev.null c b n a 2017 SKMM 2323 Mechanics of Fluids 2 Potential Flow 10 / 11

Bibliography 1 WHITE, F. M. (2011): Fluid Mechanics, 7ed, ISBN 978-0-07-352934-9, McGraw-Hill ibn.abdullah@dev.null c b n a 2017 SKMM 2323 Mechanics of Fluids 2 Potential Flow 11 / 11

... must end... and I end my presentation with two supplications my Lord! increase me in knowledge (TAA-HAA (20):114) O Allah! We ask You for knowledge that is of benefit (IBN MAJAH)