2005reading4 - 2.016 Hydrodynamics Reading#4 2.016...

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2.016 Hydrodynamics Reading #4 2.016 Hydrodynamics Prof. A.H. Techet Potential Flow Theory “When a flow is both frictionless and irrotational, pleasant things happen.” F.M. White, Fluid Mechanics 4th ed. We can treat external flows around bodies as invicid (i.e. frictionless) and irrotational (i.e. the fluid particles are not rotating). This is because the viscous effects are limited to a thin layer next to the body called the boundary layer. In graduate classes like 2.25, you’ll learn how to solve for the invicid flow and then correct this within the boundary layer by considering viscosity. For now, let’s just learn how to solve for the invicid flow. We can define a potential function , , , , as a continuous function that satisfies the ( x z t ) basic laws of fluid mechanics: conservation of mass and momentum , assuming incompressible, inviscid and irrotational flow. There is a vector identity (prove it for yourself!) that states for any scalar, , ±² ± ³ = 0 By definition, for irrotational flow , r ±² V = 0 Therefore r V = ± where ( , , ) is the velocity potential function . Such that the components of = x y , z t velocity in Cartesian coordinates, as functions of space and time, are ± ³ ± ² ± ² u dx = , v dy = and w = dz (4.1) version 1.0 updated 9/22/2005 -1- 2005 A. Techet
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2.016 Hydrodynamics Reading #4 Laplace Equation The velocity must still satisfy the conservation of mass equation. We can substitute in the relationship between potential and velocity and arrive at the Laplace Equation , which we will revisit in our discussion on linear waves. ± u + ± v + ± w = 0 (4.2) ± x ± y ± z 2 2 2 ± ² ± ² ± ² + + = 0 (4.3) 2 2 2 ± x ± y ± z LaplaceEquation ± ² 2 = 0 For your reference given below is the Laplace equation in different coordinate systems: Cartesian, cylindrical and spherical. Cartesian Coordinates ( x, y, z ) r ³ ² ³ ² ˆ ³ ² V = ui ˆ + vj ˆ + wk ˆ = i ˆ + j + k ˆ = ² ² ³ x ³ y ³ z 2 2 2 2 ³ ² ³ ² ³ ² ² ² = + + = 0 2 2 2 ³ x ³ y ³ z Cylindrical Coordinates ( r, ± , z ) r 2 = x 2 + y 2 , ² = tan ± y 1 ( ) x r ³ ² V = u e ˆ + u e ˆ + u e ˆ = ³ ² e ˆ + 1 ³ ² e ˆ + e ˆ = ³ ² r r ´ ´ z z r ´ z ³ r r ³ ´ ³ z 2 2 2 ³ ² 1 ³ ² 1 ³ 2 ² ³ ² ³ ² = + + + = 0 2 2 2 2 ³ r r ³ r r ³ ´ ³ z 1 243 4 1 ³ µ ³ ² r r ³ r ³ r version 1.0 updated 9/22/2005 -2- 2005 A. Techet
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2.016 Hydrodynamics Reading #4 version 1.0 updated 9/22/2005 -3- 2005 A. Techet Spherical Coordinates ( r, , ) 2 2 2 2 r x y z = + + , ( ) 1 cos x r = , or cos x r = , ( ) 1 tan z y = r V = u r ˆ e r + u ˆ e + u ˆ e = r ˆ e r + 1 r ˆ e + 1 r sin ˆ e = 2 = 2 r 2 + 2 r r 1 r 2 r r 2 r 1 2 4 3 4 + 1 r 2 sin sin + 1 r 2 sin 2 2 2 = 0 Potential Lines Lines of constant are called potential lines of the flow. In two dimensions d = x dx + y dy d = udx + vdy Since d = 0 along a potential line, we have dy dx = u v (4.4) Recall that streamlines are lines everywhere tangent to the velocity, dy dx = v u , so potential lines are perpendicular to the streamlines . For inviscid and irrotational flow is indeed quite pleasant to use potential function, , to represent the velocity field, as it reduced the problem from having three unknowns ( u, v, w ) to only one unknown ( ).
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